Given that lim x l2 fsxd 4 lim x l2 tsxd 22 lim x l2 hsxd 0 find the limits that exist. If the limit does not exist, explain why. (a) lim xl 2 f fsxd 1 5tsxdg (b) lim xl 2 f tsxdg3 (c) lim xl2 sfsxd (d) lim xl 2 3fsxd tsxd (e) lim xl 2 tsxd hsxd (f) lim xl 2 tsxdhsxd
Read moreTable of Contents
1
Functions and Sequences
1.1
Four Ways to Represent a Function
1.2
A Catalog of Essential Functions
1.3
New Functions from Old Functions
1.4
Exponential Functions
1.5
Logarithms; Semilog and Log-Log Plots
1.6
Sequences and Difference Equations
2
Limits
2.1
Limits of Sequences
2.2
Limits of Functions at Infinity
2.3
Limits of Functions at Finite Numbers
2.4
Limits: Algebraic Methods
2.5
Continuity
3
Derivatives
3.1
Derivatives and Rates of Change
3.2
The Derivative as a Function
3.3
Basic Differentiation Formulas
3.4
The Chain Rule
3.5
The Chain Rule
3.6
Exponential Growth and Decay
3.7
Derivatives of the Logarithmic and Inverse Tangent Functions
3.8
Linear Approximations and Taylor Polynomials
4
Applications of Derivatives
4.1
Maximum and Minimum Values
4.2
How Derivatives Affect the Shape of a Graph
4.3
LHospitals Rule: Comparing Rates of Growth
4.4
Optimization Problems
4.5
Recursions: Equilibria and Stability
4.6
Antiderivatives
5
Integrals
5.1
Areas, Distances, and Pathogenesis
5.2
The Definite Integral
5.3
The Fundamental Theorem of Calculus
5.4
The Substitution Rule
5.5
Integration by Parts
5.6
Partial Fractions
5.7
Integration Using Tables and Computer Algebra Systems
5.8
Improper Integrals
6
Applications of Integrals
6.1
Areas Between Curves
6.2
Average Values
6.3
Further Applications to Biology
6.4
Volumes
7
Differential Equations
7.1
Modeling with Differential Equations
7.2
Phase Plots, Equilibria, and Stability
7.3
Direction Fields and Eulers Method
7.4
Separable Equations
7.5
Phase Plane Analysis
7.6
Phase Plane Analysis
8
Vectors and Matrix Models
8.1
Coordinate Systems
8.2
Vectors
8.3
The Dot Product
8.4
Matrix Algebra
8.5
Matrices and the Dynamics of Vectors
8.6
Eigenvectors and Eigenvalues
8.7
Eigenvectors and Eigenvalues
8.8
Iterated Matrix Models
9
Multivariable Calculus
9.1
Functions of Several Variables
9.2
Partial Derivatives
9.3
Tangent Planes and Linear Approximations
9.4
The Chain Rule
9.5
Directional Derivatives and the Gradient Vector
9.6
Maximum and Minimum Values
10
Systems of Linear Differential Equations
10.1
Qualitative Analysis of Linear Systems
10.2
Qualitative Analysis of Linear Systems
10.3
Applications
10.4
Systems of Nonlinear Differential Equations
Textbook Solutions for Biocalculus: Calculus for Life Sciences
Chapter 2.4 Problem 1
Question
Given that lim x l2 fsxd 4 lim x l2 tsxd 22 lim x l2 hsxd 0 find the limits that exist. If the limit does not exist, explain why. (a) lim xl 2 f fsxd 1 5tsxdg (b) lim xl 2 f tsxdg3 (c) lim xl2 sfsxd (d) lim xl 2 3fsxd tsxd (e) lim xl 2 tsxd hsxd (f) lim xl 2 tsxdhsxd
Solution
The first step in solving 2.4 problem number 1 trying to solve the problem we have to refer to the textbook question: Given that lim x l2 fsxd 4 lim x l2 tsxd 22 lim x l2 hsxd 0 find the limits that exist. If the limit does not exist, explain why. (a) lim xl 2 f fsxd 1 5tsxdg (b) lim xl 2 f tsxdg3 (c) lim xl2 sfsxd (d) lim xl 2 3fsxd tsxd (e) lim xl 2 tsxd hsxd (f) lim xl 2 tsxdhsxd
From the textbook chapter Limits: Algebraic Methods you will find a few key concepts needed to solve this.
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Title
Biocalculus: Calculus for Life Sciences 1
Author
James Stewart, Troy Day
ISBN
9781133109631
Given that lim x l2 fsxd 4 lim x l2 tsxd 22 lim x l2 hsxd 0 find the limits that exist
Chapter 2.4 textbook questions
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Chapter 2: Problem 1 Biocalculus: Calculus for Life Sciences 1
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Chapter 2: Problem 2 Biocalculus: Calculus for Life Sciences 1
The graphs of f and t are given. Use them to evaluate each limit, if it exists. If the limit does not exist, explain why. (a) lim xl 2 f fsxd 1 tsxdg (b) lim xl1 f fsxd 1 tsxdg (c) lim xl 0 f fsxd tsxdg (d) lim xl21 fsxd tsxd (e) lim xl 2 fx 3 fsxdg (f) lim xl1 s3 1 fsxd
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Chapter 2: Problem 3 Biocalculus: Calculus for Life Sciences 1
37 Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). 3. lim xl22 s3x 4 1 2x 2 2 x 1 1d 4. lim tl21 st 2 1 1d 3 st 1 3d 5 5. lim xl2 2x 2 1 1 3x 2 2 6. lim x l0 cos4 x 5 1 2x 3 7. lim l y2 sin
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Chapter 2: Problem 4 Biocalculus: Calculus for Life Sciences 1
37 Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). 3. lim xl22 s3x 4 1 2x 2 2 x 1 1d 4. lim tl21 st 2 1 1d 3 st 1 3d 5 5. lim xl2 2x 2 1 1 3x 2 2 6. lim x l0 cos4 x 5 1 2x 3 7. lim l y2 sin
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Chapter 2: Problem 5 Biocalculus: Calculus for Life Sciences 1
37 Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). 3. lim xl22 s3x 4 1 2x 2 2 x 1 1d 4. lim tl21 st 2 1 1d 3 st 1 3d 5 5. lim xl2 2x 2 1 1 3x 2 2 6. lim x l0 cos4 x 5 1 2x 3 7. lim l y2 sin
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Chapter 2: Problem 6 Biocalculus: Calculus for Life Sciences 1
37 Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). 3. lim xl22 s3x 4 1 2x 2 2 x 1 1d 4. lim tl21 st 2 1 1d 3 st 1 3d 5 5. lim xl2 2x 2 1 1 3x 2 2 6. lim x l0 cos4 x 5 1 2x 3 7. lim l y2 sin
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Chapter 2: Problem 7 Biocalculus: Calculus for Life Sciences 1
37 Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). 3. lim xl22 s3x 4 1 2x 2 2 x 1 1d 4. lim tl21 st 2 1 1d 3 st 1 3d 5 5. lim xl2 2x 2 1 1 3x 2 2 6. lim x l0 cos4 x 5 1 2x 3 7. lim l y2 sin
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Chapter 2: Problem 8 Biocalculus: Calculus for Life Sciences 1
(a) What is wrong with the following equation? x 2 1 x 2 6 x 2 2 x 1 3 (b) In view of part (a), explain why the equation lim xl 2 x 2 1 x 2 6 x 2 2 lim xl 2 sx 1 3d is correct
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Chapter 2: Problem 9 Biocalculus: Calculus for Life Sciences 1
924 Evaluate the limit, if it exists. 9. lim xl 5 x 2 2 6x 1 5 x 2 5 10. lim xl 4 x 2 2 4x x 2 2 3x 2 4 11. lim xl 5 x 2 2 5x 1 6 x 2 5 12. lim xl21 2x 2 1 3x 1 1 x 2 2 2x 2 3 13. lim tl23 t 2 2 9 2t 2 1 7t 1 3 14. lim xl21 x 2 2 4x x 2 2 3x 2 4 15. lim hl0 s4 1 hd 2 2 16 h 16. lim hl 0 s2 1 hd 3 2 8 h 17. lim xl22 x 1 2 x 3 1 8 18. lim hl 0 s1 1 h 2 1 h 19. lim xl24 1 4 1 1 x 4 1 x 20. lim xl21 x 2 1 2x 1 1 x 4 2 1 21. lim xl16 4 2 sx 16x 2 x 2 22. lim tl 0 S 1 t 2 1 t 2 1 t D 23. lim tl0 S 1 ts1 1 t 2 1 t D 24. lim x
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Chapter 2: Problem 10 Biocalculus: Calculus for Life Sciences 1
924 Evaluate the limit, if it exists. 9. lim xl 5 x 2 2 6x 1 5 x 2 5 10. lim xl 4 x 2 2 4x x 2 2 3x 2 4 11. lim xl 5 x 2 2 5x 1 6 x 2 5 12. lim xl21 2x 2 1 3x 1 1 x 2 2 2x 2 3 13. lim tl23 t 2 2 9 2t 2 1 7t 1 3 14. lim xl21 x 2 2 4x x 2 2 3x 2 4 15. lim hl0 s4 1 hd 2 2 16 h 16. lim hl 0 s2 1 hd 3 2 8 h 17. lim xl22 x 1 2 x 3 1 8 18. lim hl 0 s1 1 h 2 1 h 19. lim xl24 1 4 1 1 x 4 1 x 20. lim xl21 x 2 1 2x 1 1 x 4 2 1 21. lim xl16 4 2 sx 16x 2 x 2 22. lim tl 0 S 1 t 2 1 t 2 1 t D 23. lim tl0 S 1 ts1 1 t 2 1 t D 24. lim x
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Chapter 2: Problem 11 Biocalculus: Calculus for Life Sciences 1
924 Evaluate the limit, if it exists. 9. lim xl 5 x 2 2 6x 1 5 x 2 5 10. lim xl 4 x 2 2 4x x 2 2 3x 2 4 11. lim xl 5 x 2 2 5x 1 6 x 2 5 12. lim xl21 2x 2 1 3x 1 1 x 2 2 2x 2 3 13. lim tl23 t 2 2 9 2t 2 1 7t 1 3 14. lim xl21 x 2 2 4x x 2 2 3x 2 4 15. lim hl0 s4 1 hd 2 2 16 h 16. lim hl 0 s2 1 hd 3 2 8 h 17. lim xl22 x 1 2 x 3 1 8 18. lim hl 0 s1 1 h 2 1 h 19. lim xl24 1 4 1 1 x 4 1 x 20. lim xl21 x 2 1 2x 1 1 x 4 2 1 21. lim xl16 4 2 sx 16x 2 x 2 22. lim tl 0 S 1 t 2 1 t 2 1 t D 23. lim tl0 S 1 ts1 1 t 2 1 t D 24. lim x
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Chapter 2: Problem 12 Biocalculus: Calculus for Life Sciences 1
924 Evaluate the limit, if it exists. 9. lim xl 5 x 2 2 6x 1 5 x 2 5 10. lim xl 4 x 2 2 4x x 2 2 3x 2 4 11. lim xl 5 x 2 2 5x 1 6 x 2 5 12. lim xl21 2x 2 1 3x 1 1 x 2 2 2x 2 3 13. lim tl23 t 2 2 9 2t 2 1 7t 1 3 14. lim xl21 x 2 2 4x x 2 2 3x 2 4 15. lim hl0 s4 1 hd 2 2 16 h 16. lim hl 0 s2 1 hd 3 2 8 h 17. lim xl22 x 1 2 x 3 1 8 18. lim hl 0 s1 1 h 2 1 h 19. lim xl24 1 4 1 1 x 4 1 x 20. lim xl21 x 2 1 2x 1 1 x 4 2 1 21. lim xl16 4 2 sx 16x 2 x 2 22. lim tl 0 S 1 t 2 1 t 2 1 t D 23. lim tl0 S 1 ts1 1 t 2 1 t D 24. lim x
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Chapter 2: Problem 13 Biocalculus: Calculus for Life Sciences 1
924 Evaluate the limit, if it exists. 9. lim xl 5 x 2 2 6x 1 5 x 2 5 10. lim xl 4 x 2 2 4x x 2 2 3x 2 4 11. lim xl 5 x 2 2 5x 1 6 x 2 5 12. lim xl21 2x 2 1 3x 1 1 x 2 2 2x 2 3 13. lim tl23 t 2 2 9 2t 2 1 7t 1 3 14. lim xl21 x 2 2 4x x 2 2 3x 2 4 15. lim hl0 s4 1 hd 2 2 16 h 16. lim hl 0 s2 1 hd 3 2 8 h 17. lim xl22 x 1 2 x 3 1 8 18. lim hl 0 s1 1 h 2 1 h 19. lim xl24 1 4 1 1 x 4 1 x 20. lim xl21 x 2 1 2x 1 1 x 4 2 1 21. lim xl16 4 2 sx 16x 2 x 2 22. lim tl 0 S 1 t 2 1 t 2 1 t D 23. lim tl0 S 1 ts1 1 t 2 1 t D 24. lim x
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Chapter 2: Problem 14 Biocalculus: Calculus for Life Sciences 1
924 Evaluate the limit, if it exists. 9. lim xl 5 x 2 2 6x 1 5 x 2 5 10. lim xl 4 x 2 2 4x x 2 2 3x 2 4 11. lim xl 5 x 2 2 5x 1 6 x 2 5 12. lim xl21 2x 2 1 3x 1 1 x 2 2 2x 2 3 13. lim tl23 t 2 2 9 2t 2 1 7t 1 3 14. lim xl21 x 2 2 4x x 2 2 3x 2 4 15. lim hl0 s4 1 hd 2 2 16 h 16. lim hl 0 s2 1 hd 3 2 8 h 17. lim xl22 x 1 2 x 3 1 8 18. lim hl 0 s1 1 h 2 1 h 19. lim xl24 1 4 1 1 x 4 1 x 20. lim xl21 x 2 1 2x 1 1 x 4 2 1 21. lim xl16 4 2 sx 16x 2 x 2 22. lim tl 0 S 1 t 2 1 t 2 1 t D 23. lim tl0 S 1 ts1 1 t 2 1 t D 24. lim x
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Chapter 2: Problem 15 Biocalculus: Calculus for Life Sciences 1
924 Evaluate the limit, if it exists. 9. lim xl 5 x 2 2 6x 1 5 x 2 5 10. lim xl 4 x 2 2 4x x 2 2 3x 2 4 11. lim xl 5 x 2 2 5x 1 6 x 2 5 12. lim xl21 2x 2 1 3x 1 1 x 2 2 2x 2 3 13. lim tl23 t 2 2 9 2t 2 1 7t 1 3 14. lim xl21 x 2 2 4x x 2 2 3x 2 4 15. lim hl0 s4 1 hd 2 2 16 h 16. lim hl 0 s2 1 hd 3 2 8 h 17. lim xl22 x 1 2 x 3 1 8 18. lim hl 0 s1 1 h 2 1 h 19. lim xl24 1 4 1 1 x 4 1 x 20. lim xl21 x 2 1 2x 1 1 x 4 2 1 21. lim xl16 4 2 sx 16x 2 x 2 22. lim tl 0 S 1 t 2 1 t 2 1 t D 23. lim tl0 S 1 ts1 1 t 2 1 t D 24. lim x
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Chapter 2: Problem 16 Biocalculus: Calculus for Life Sciences 1
924 Evaluate the limit, if it exists. 9. lim xl 5 x 2 2 6x 1 5 x 2 5 10. lim xl 4 x 2 2 4x x 2 2 3x 2 4 11. lim xl 5 x 2 2 5x 1 6 x 2 5 12. lim xl21 2x 2 1 3x 1 1 x 2 2 2x 2 3 13. lim tl23 t 2 2 9 2t 2 1 7t 1 3 14. lim xl21 x 2 2 4x x 2 2 3x 2 4 15. lim hl0 s4 1 hd 2 2 16 h 16. lim hl 0 s2 1 hd 3 2 8 h 17. lim xl22 x 1 2 x 3 1 8 18. lim hl 0 s1 1 h 2 1 h 19. lim xl24 1 4 1 1 x 4 1 x 20. lim xl21 x 2 1 2x 1 1 x 4 2 1 21. lim xl16 4 2 sx 16x 2 x 2 22. lim tl 0 S 1 t 2 1 t 2 1 t D 23. lim tl0 S 1 ts1 1 t 2 1 t D 24. lim x
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Chapter 2: Problem 17 Biocalculus: Calculus for Life Sciences 1
924 Evaluate the limit, if it exists. 9. lim xl 5 x 2 2 6x 1 5 x 2 5 10. lim xl 4 x 2 2 4x x 2 2 3x 2 4 11. lim xl 5 x 2 2 5x 1 6 x 2 5 12. lim xl21 2x 2 1 3x 1 1 x 2 2 2x 2 3 13. lim tl23 t 2 2 9 2t 2 1 7t 1 3 14. lim xl21 x 2 2 4x x 2 2 3x 2 4 15. lim hl0 s4 1 hd 2 2 16 h 16. lim hl 0 s2 1 hd 3 2 8 h 17. lim xl22 x 1 2 x 3 1 8 18. lim hl 0 s1 1 h 2 1 h 19. lim xl24 1 4 1 1 x 4 1 x 20. lim xl21 x 2 1 2x 1 1 x 4 2 1 21. lim xl16 4 2 sx 16x 2 x 2 22. lim tl 0 S 1 t 2 1 t 2 1 t D 23. lim tl0 S 1 ts1 1 t 2 1 t D 24. lim x
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Chapter 2: Problem 18 Biocalculus: Calculus for Life Sciences 1
924 Evaluate the limit, if it exists. 9. lim xl 5 x 2 2 6x 1 5 x 2 5 10. lim xl 4 x 2 2 4x x 2 2 3x 2 4 11. lim xl 5 x 2 2 5x 1 6 x 2 5 12. lim xl21 2x 2 1 3x 1 1 x 2 2 2x 2 3 13. lim tl23 t 2 2 9 2t 2 1 7t 1 3 14. lim xl21 x 2 2 4x x 2 2 3x 2 4 15. lim hl0 s4 1 hd 2 2 16 h 16. lim hl 0 s2 1 hd 3 2 8 h 17. lim xl22 x 1 2 x 3 1 8 18. lim hl 0 s1 1 h 2 1 h 19. lim xl24 1 4 1 1 x 4 1 x 20. lim xl21 x 2 1 2x 1 1 x 4 2 1 21. lim xl16 4 2 sx 16x 2 x 2 22. lim tl 0 S 1 t 2 1 t 2 1 t D 23. lim tl0 S 1 ts1 1 t 2 1 t D 24. lim x
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Chapter 2: Problem 19 Biocalculus: Calculus for Life Sciences 1
924 Evaluate the limit, if it exists. 9. lim xl 5 x 2 2 6x 1 5 x 2 5 10. lim xl 4 x 2 2 4x x 2 2 3x 2 4 11. lim xl 5 x 2 2 5x 1 6 x 2 5 12. lim xl21 2x 2 1 3x 1 1 x 2 2 2x 2 3 13. lim tl23 t 2 2 9 2t 2 1 7t 1 3 14. lim xl21 x 2 2 4x x 2 2 3x 2 4 15. lim hl0 s4 1 hd 2 2 16 h 16. lim hl 0 s2 1 hd 3 2 8 h 17. lim xl22 x 1 2 x 3 1 8 18. lim hl 0 s1 1 h 2 1 h 19. lim xl24 1 4 1 1 x 4 1 x 20. lim xl21 x 2 1 2x 1 1 x 4 2 1 21. lim xl16 4 2 sx 16x 2 x 2 22. lim tl 0 S 1 t 2 1 t 2 1 t D 23. lim tl0 S 1 ts1 1 t 2 1 t D 24. lim x
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Chapter 2: Problem 20 Biocalculus: Calculus for Life Sciences 1
924 Evaluate the limit, if it exists. 9. lim xl 5 x 2 2 6x 1 5 x 2 5 10. lim xl 4 x 2 2 4x x 2 2 3x 2 4 11. lim xl 5 x 2 2 5x 1 6 x 2 5 12. lim xl21 2x 2 1 3x 1 1 x 2 2 2x 2 3 13. lim tl23 t 2 2 9 2t 2 1 7t 1 3 14. lim xl21 x 2 2 4x x 2 2 3x 2 4 15. lim hl0 s4 1 hd 2 2 16 h 16. lim hl 0 s2 1 hd 3 2 8 h 17. lim xl22 x 1 2 x 3 1 8 18. lim hl 0 s1 1 h 2 1 h 19. lim xl24 1 4 1 1 x 4 1 x 20. lim xl21 x 2 1 2x 1 1 x 4 2 1 21. lim xl16 4 2 sx 16x 2 x 2 22. lim tl 0 S 1 t 2 1 t 2 1 t D 23. lim tl0 S 1 ts1 1 t 2 1 t D 24. lim x
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Chapter 2: Problem 21 Biocalculus: Calculus for Life Sciences 1
924 Evaluate the limit, if it exists. 9. lim xl 5 x 2 2 6x 1 5 x 2 5 10. lim xl 4 x 2 2 4x x 2 2 3x 2 4 11. lim xl 5 x 2 2 5x 1 6 x 2 5 12. lim xl21 2x 2 1 3x 1 1 x 2 2 2x 2 3 13. lim tl23 t 2 2 9 2t 2 1 7t 1 3 14. lim xl21 x 2 2 4x x 2 2 3x 2 4 15. lim hl0 s4 1 hd 2 2 16 h 16. lim hl 0 s2 1 hd 3 2 8 h 17. lim xl22 x 1 2 x 3 1 8 18. lim hl 0 s1 1 h 2 1 h 19. lim xl24 1 4 1 1 x 4 1 x 20. lim xl21 x 2 1 2x 1 1 x 4 2 1 21. lim xl16 4 2 sx 16x 2 x 2 22. lim tl 0 S 1 t 2 1 t 2 1 t D 23. lim tl0 S 1 ts1 1 t 2 1 t D 24. lim x
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Chapter 2: Problem 22 Biocalculus: Calculus for Life Sciences 1
924 Evaluate the limit, if it exists. 9. lim xl 5 x 2 2 6x 1 5 x 2 5 10. lim xl 4 x 2 2 4x x 2 2 3x 2 4 11. lim xl 5 x 2 2 5x 1 6 x 2 5 12. lim xl21 2x 2 1 3x 1 1 x 2 2 2x 2 3 13. lim tl23 t 2 2 9 2t 2 1 7t 1 3 14. lim xl21 x 2 2 4x x 2 2 3x 2 4 15. lim hl0 s4 1 hd 2 2 16 h 16. lim hl 0 s2 1 hd 3 2 8 h 17. lim xl22 x 1 2 x 3 1 8 18. lim hl 0 s1 1 h 2 1 h 19. lim xl24 1 4 1 1 x 4 1 x 20. lim xl21 x 2 1 2x 1 1 x 4 2 1 21. lim xl16 4 2 sx 16x 2 x 2 22. lim tl 0 S 1 t 2 1 t 2 1 t D 23. lim tl0 S 1 ts1 1 t 2 1 t D 24. lim x
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Chapter 2: Problem 23 Biocalculus: Calculus for Life Sciences 1
924 Evaluate the limit, if it exists. 9. lim xl 5 x 2 2 6x 1 5 x 2 5 10. lim xl 4 x 2 2 4x x 2 2 3x 2 4 11. lim xl 5 x 2 2 5x 1 6 x 2 5 12. lim xl21 2x 2 1 3x 1 1 x 2 2 2x 2 3 13. lim tl23 t 2 2 9 2t 2 1 7t 1 3 14. lim xl21 x 2 2 4x x 2 2 3x 2 4 15. lim hl0 s4 1 hd 2 2 16 h 16. lim hl 0 s2 1 hd 3 2 8 h 17. lim xl22 x 1 2 x 3 1 8 18. lim hl 0 s1 1 h 2 1 h 19. lim xl24 1 4 1 1 x 4 1 x 20. lim xl21 x 2 1 2x 1 1 x 4 2 1 21. lim xl16 4 2 sx 16x 2 x 2 22. lim tl 0 S 1 t 2 1 t 2 1 t D 23. lim tl0 S 1 ts1 1 t 2 1 t D 24. lim x
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Chapter 2: Problem 24 Biocalculus: Calculus for Life Sciences 1
924 Evaluate the limit, if it exists. 9. lim xl 5 x 2 2 6x 1 5 x 2 5 10. lim xl 4 x 2 2 4x x 2 2 3x 2 4 11. lim xl 5 x 2 2 5x 1 6 x 2 5 12. lim xl21 2x 2 1 3x 1 1 x 2 2 2x 2 3 13. lim tl23 t 2 2 9 2t 2 1 7t 1 3 14. lim xl21 x 2 2 4x x 2 2 3x 2 4 15. lim hl0 s4 1 hd 2 2 16 h 16. lim hl 0 s2 1 hd 3 2 8 h 17. lim xl22 x 1 2 x 3 1 8 18. lim hl 0 s1 1 h 2 1 h 19. lim xl24 1 4 1 1 x 4 1 x 20. lim xl21 x 2 1 2x 1 1 x 4 2 1 21. lim xl16 4 2 sx 16x 2 x 2 22. lim tl 0 S 1 t 2 1 t 2 1 t D 23. lim tl0 S 1 ts1 1 t 2 1 t D 24. lim x
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Chapter 2: Problem 25 Biocalculus: Calculus for Life Sciences 1
(a) Estimate the value of lim xl 0 x s1 1 3x 2 1 by graphing the function fsxd xyss1 1 3x 2 1d. (b) Make a table of values of fsxd for x close to 0 and guess the value of the limit. (c) Use the Limit Laws to prove that your guess is corr
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Chapter 2: Problem 26 Biocalculus: Calculus for Life Sciences 1
(a) Use a graph of fsxd s3 1 x 2 s3 x to estimate the value of limxl0 fsxd to two decimal places. (b) Use a table of values of fsxd to estimate the limit to four decimal places. (c) Use the Limit Laws to find the exact value of the limit.
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Chapter 2: Problem 27 Biocalculus: Calculus for Life Sciences 1
Use the Squeeze Theorem to show that limxl0 sx 2 cos 20 xd 0. Illustrate by graphing the functions fsxd 2x 2 , tsxd x 2 cos 20 x, and hsxd x 2 on the same screen.
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Chapter 2: Problem 28 Biocalculus: Calculus for Life Sciences 1
Use the Squeeze Theorem to show that lim xl 0 sx 3 1 x 2 sin x 0 Illustrate by graphing the functions f, t, and h (in the notation of the Squeeze Theorem) on the same screen.
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Chapter 2: Problem 29 Biocalculus: Calculus for Life Sciences 1
If 4x 2 9 < fsxd < x 2 2 4x 1 7 for x > 0, find lim xl 4 fsxd.
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Chapter 2: Problem 30 Biocalculus: Calculus for Life Sciences 1
If 2x < tsxd < x 4 2 x 2 1 2 for all x, evaluate lim xl1 tsxd.
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Chapter 2: Problem 31 Biocalculus: Calculus for Life Sciences 1
Prove that limxl 0 x 4 cos 2 x 0.
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Chapter 2: Problem 32 Biocalculus: Calculus for Life Sciences 1
Gene regulation Genes produce molecules called mRNA that go on to produce proteins. High concentrations of protein inhibit the production of mRNA, leading to stable gene regulation. This process has been modeled (see Section 10.3) to show that the concentration of mRNA over time is given by the equation mstd 1 2e2t ssin t 2 cos td 1 1 2 (a) Evaluate limtl0 mstd and interpret your result. (b) Use the Squeeze Theorem to evaluate limtl` mstd and interpret your result.
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Chapter 2: Problem 33 Biocalculus: Calculus for Life Sciences 1
3336 Find the limit, if it exists. If the limit does not exist, explain why. 33. lim xl3 s2x 1 | x 2 3 |d 34. lim xl26 2x 1 12 | x 1 6 | 35. lim xl 02 S 1 x 2 1 | x | D 36. lim xl22 2 2 | x | 2 1 x
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Chapter 2: Problem 34 Biocalculus: Calculus for Life Sciences 1
3336 Find the limit, if it exists. If the limit does not exist, explain why. 33. lim xl3 s2x 1 | x 2 3 |d 34. lim xl26 2x 1 12 | x 1 6 | 35. lim xl 02 S 1 x 2 1 | x | D 36. lim xl22 2 2 | x | 2 1 x
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Chapter 2: Problem 35 Biocalculus: Calculus for Life Sciences 1
3336 Find the limit, if it exists. If the limit does not exist, explain why. 33. lim xl3 s2x 1 | x 2 3 |d 34. lim xl26 2x 1 12 | x 1 6 | 35. lim xl 02 S 1 x 2 1 | x | D 36. lim xl22 2 2 | x | 2 1 x
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Chapter 2: Problem 36 Biocalculus: Calculus for Life Sciences 1
3336 Find the limit, if it exists. If the limit does not exist, explain why. 33. lim xl3 s2x 1 | x 2 3 |d 34. lim xl26 2x 1 12 | x 1 6 | 35. lim xl 02 S 1 x 2 1 | x | D 36. lim xl22 2 2 | x | 2 1 x
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Chapter 2: Problem 37 Biocalculus: Calculus for Life Sciences 1
Let tsxd x 2 1 x 2 6 | x 2 2 | . (a) Find (i) lim xl 21 tsxd (ii) lim xl 22 tsxd (b) Does limxl2 tsxd exist? (c) Sketch the graph of t.
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Chapter 2: Problem 38 Biocalculus: Calculus for Life Sciences 1
Let fsxd H x 2 1 1 sx 2 2d 2 if x , 1 if x > 1 (a) Find limx l12 fsxd and limx l1 1 fsxd. (b) Does lim x l1 fsxd exist? (c) Sketch the graph of f
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Chapter 2: Problem 39 Biocalculus: Calculus for Life Sciences 1
3944 Find the limit. 39. lim xl0 sin 3x x 40. lim xl0 sin 4x sin 6x 41. lim tl0 tan 6t sin 2t 42. lim tl 0 sin2 3t t 2 43. lim l0 sin 1 tan 44. lim x l0 x cot x
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Chapter 2: Problem 40 Biocalculus: Calculus for Life Sciences 1
3944 Find the limit. 39. lim xl0 sin 3x x 40. lim xl0 sin 4x sin 6x 41. lim tl0 tan 6t sin 2t 42. lim tl 0 sin2 3t t 2 43. lim l0 sin 1 tan 44. lim x l0 x cot x
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Chapter 2: Problem 41 Biocalculus: Calculus for Life Sciences 1
3944 Find the limit. 39. lim xl0 sin 3x x 40. lim xl0 sin 4x sin 6x 41. lim tl0 tan 6t sin 2t 42. lim tl 0 sin2 3t t 2 43. lim l0 sin 1 tan 44. lim x l0 x cot x
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Chapter 2: Problem 42 Biocalculus: Calculus for Life Sciences 1
3944 Find the limit. 39. lim xl0 sin 3x x 40. lim xl0 sin 4x sin 6x 41. lim tl0 tan 6t sin 2t 42. lim tl 0 sin2 3t t 2 43. lim l0 sin 1 tan 44. lim x l0 x cot x
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Chapter 2: Problem 43 Biocalculus: Calculus for Life Sciences 1
3944 Find the limit. 39. lim xl0 sin 3x x 40. lim xl0 sin 4x sin 6x 41. lim tl0 tan 6t sin 2t 42. lim tl 0 sin2 3t t 2 43. lim l0 sin 1 tan 44. lim x l0 x cot x
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Chapter 2: Problem 44 Biocalculus: Calculus for Life Sciences 1
3944 Find the limit. 39. lim xl0 sin 3x x 40. lim xl0 sin 4x sin 6x 41. lim tl0 tan 6t sin 2t 42. lim tl 0 sin2 3t t 2 43. lim l0 sin 1 tan 44. lim x l0 x cot x
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Chapter 2: Problem 45 Biocalculus: Calculus for Life Sciences 1
(a) If p is a polynomial, show that limxla psxd psad. (b) If r is a rational function, use part (a) to show that limxla rsxd rsad for every number a in the domain of
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Chapter 2: Problem 46 Biocalculus: Calculus for Life Sciences 1
In the theory of relativity, the Lorentz contraction formula L L0s1 2 v 2 yc 2 expresses the length L of an object as a function of its velocity v with respect to an observer, where L0 is the length of the object at rest and c is the speed of light. Find limvlc2 L and interpret the result. Why is a left-hand limit necessary?
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Chapter 2: Problem 47 Biocalculus: Calculus for Life Sciences 1
To prove that sine has the Direct Substitution Property we need to show that limxla sin x sin a for every real number a. If we let h x 2 a, then x a 1 h and x l a &? h l 0. So an equivalent statement is that lim hl0 sinsa 1 hd sin a Use (5) to show that this is true.
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Chapter 2: Problem 48 Biocalculus: Calculus for Life Sciences 1
Prove that cosine has the Direct Substitution Property.
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Chapter 2: Problem 49 Biocalculus: Calculus for Life Sciences 1
If lim xl1 fsxd 2 8 x 2 1 10, find lim xl1 fsx
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Chapter 2: Problem 50 Biocalculus: Calculus for Life Sciences 1
If limxl 0 fsxd x 2 5, find the following limits. (a) lim x l0 fsxd (b) lim xl 0 fsx
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Chapter 2: Problem 51 Biocalculus: Calculus for Life Sciences 1
Is there a number a such that lim xl22 3x 2 1 ax 1 a 1 3 x 2 1 x 2 2 exists? If so, find the value of a and the value of the li
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Chapter 2: Problem 52 Biocalculus: Calculus for Life Sciences 1
The figure shows a fixed circle C1 with equation sx 2 1d 2 1 y 2 1 and a shrinking circle C2 with radius r and center the origin. P is the point s0, rd, Q is the upper point of intersection of the two circles, and R is the point of intersection of the line PQ and the x-axis. What happens to R as C2 shrinks, that is, as r l 01?
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