7076. Useful factorization formula Calculate

How is lim xSa f 1x2 calculated if f is a polynomial function?

How are lim xSa- f 1x2 and lim xSa+ f 1x2 calculated if f is a polynomial function?

For what values of a does lim xSa r1x2 = r1a2 if r is a rational function?

Assume lim xS3 g1x2 = 4 and f 1x2 = g1x2 whenever x _ 3. Evaluate lim xS3 f 1x2, if possible.

Explain why lim xS3 x2 - 7x + 12 x - 3 = lim xS3 1x - 42.

If lim xS2 f 1x2 = -8, find lim xS2 1 f 1x222>3.

Suppose p and q are polynomials. If lim xS0 p1x2 q1x2 = 10 and q102 = 2, find p102.

Suppose lim xS2 f 1x2 = lim xS2 h1x2 = 5. Find lim xS2 g1x2, where f 1x2 g1x2 h1x2, for all x.

Evaluate lim xS5 2x2 - 9.

Suppose f 1x2 = b 4 if x 3 x + 2 if x 7 3. Compute lim xS3- f 1x2 and lim xS3+ f 1x2.

1116. Limits of linear functions Evaluate the following limits. lim xS4 13x - 72

1116. Limits of linear functions Evaluate the following limits. lim xS1 1-2x + 52

1116. Limits of linear functions Evaluate the following limits. lim xS-9 5x

1116. Limits of linear functions Evaluate the following limits. lim xS2 1-3x2

1116. Limits of linear functions Evaluate the following limits. lim xS6 4

1116. Limits of linear functions Evaluate the following limits. lim xS-5 p

1724. Applying limit laws Assume lim xS1 f 1x2 = 8, lim xS1 g1x2 = 3, and lim xS1 h1x2 = 2. Compute the following limits and state the limit laws used to justify your computations. lim xS1 14f 1x22

1724. Applying limit laws Assume lim xS1 f 1x2 = 8, lim xS1 g1x2 = 3, and lim xS1 h1x2 = 2. Compute the following limits and state the limit laws used to justify your computations. lim xS1 f 1x2 h1x2

1724. Applying limit laws Assume lim xS1 f 1x2 = 8, lim xS1 g1x2 = 3, and lim xS1 h1x2 = 2. Compute the following limits and state the limit laws used to justify your computations. lim xS1 1f 1x2 - g1x22

1724. Applying limit laws Assume lim xS1 f 1x2 = 8, lim xS1 g1x2 = 3, and lim xS1 h1x2 = 2. Compute the following limits and state the limit laws used to justify your computations. lim xS1 1f 1x2h1x22

1724. Applying limit laws Assume lim xS1 f 1x2 = 8, lim xS1 g1x2 = 3, and lim xS1 h1x2 = 2. Compute the following limits and state the limit laws used to justify your computations. lim xS1 f 1x2g1x2 h1x2

1724. Applying limit laws Assume lim xS1 f 1x2 = 8, lim xS1 g1x2 = 3, and lim xS1 h1x2 = 2. Compute the following limits and state the limit laws used to justify your computations. lim xS1 f 1x2 g1x2 - h1x2

1724. Applying limit laws Assume lim xS1 f 1x2 = 8, lim xS1 g1x2 = 3, and lim xS1 h1x2 = 2. Compute the following limits and state the limit laws used to justify your computations. lim xS1 1h1x225

1724. Applying limit laws Assume lim xS1 f 1x2 = 8, lim xS1 g1x2 = 3, and lim xS1 h1x2 = 2. Compute the following limits and state the limit laws used to justify your computations. lim xS1 23 f 1x2g1x2 + 3

2532. Evaluating limits Evaluate the following limits. lim xS1 12x3 - 3x2 + 4x + 52

2532. Evaluating limits Evaluate the following limits. lim tS-2 1t2 + 5t + 72

2532. Evaluating limits Evaluate the following limits. lim xS1 5x2 + 6x + 1 8x - 4

2532. Evaluating limits Evaluate the following limits. lim tS3 23 t2 - 10

2532. Evaluating limits Evaluate the following limits. lim bS2 3b 14b + 1 - 1

2532. Evaluating limits Evaluate the following limits. lim xS2 1x2 - x25

2532. Evaluating limits Evaluate the following limits. lim xS3 -5x 14x - 3

2532. Evaluating limits Evaluate the following limits. lim hS0 3 116 + 3h + 4

One-sided limits Let f 1x2 = b x2 + 1 if x 6 -1 1x + 1 if x -1. Compute the following limits or state that they do not exist. a. lim xS-1- f 1x2 b. lim xS-1+ f 1x2 c. lim xS-1 f 1x2

One-sided limits Let f 1x2 = c 0 if x -5 225 - x2 if -5 6 x 6 5 3x if x 5. Compute the following limits or state that they do not exist. a. lim xS-5- f 1x2 b. lim xS-5+ f 1x2 c. lim xS-5 f 1x2 d. lim xS5- f 1x2 e. lim xS5+ f 1x2 f. lim xS5 f 1x2

One-sided limits a. Evaluate lim xS2+ 1x - 2. b. Explain why lim xS2- 1x - 2 does not exist.

One-sided limits a. Evaluate lim xS3- A x - 3 2 - x . b. Explain why lim xS3+ A x - 3 2 - x does not exist.

Absolute value limit Show that lim xS0 _ x _ = 0 by first evaluating lim xS0- _ x _ and lim xS0+ _ x _. Recall that _ x _ = b x if x 0 -x if x 6 0.

Absolute value limit Show that lim xSa _ x _ = _ a _, for any real number. (Hint: Consider the cases a 6 0 and a 0.)

3952. Other techniques Evaluate the following limits, where a and b are fixed real numbers. lim xS1 x2 - 1 x - 1

3952. Other techniques Evaluate the following limits, where a and b are fixed real numbers. lim xS3 x2 - 2x - 3 x - 3

3952. Other techniques Evaluate the following limits, where a and b are fixed real numbers. lim xS4 x2 - 16 4 - x

3952. Other techniques Evaluate the following limits, where a and b are fixed real numbers. lim tS2 3t2 - 7t + 2 2 - t

3952. Other techniques Evaluate the following limits, where a and b are fixed real numbers. lim xSb 1x - b250 - x + b x - b

3952. Other techniques Evaluate the following limits, where a and b are fixed real numbers. lim xS-b 1x + b27+1x + b210 41x + b2

3952. Other techniques Evaluate the following limits, where a and b are fixed real numbers. lim xS-1 12x - 122 - 9 x + 1

3952. Other techniques Evaluate the following limits, where a and b are fixed real numbers. lim hS0 1 5 + h - 15 h

3952. Other techniques Evaluate the following limits, where a and b are fixed real numbers. lim xS9 1x - 3 x - 9

3952. Other techniques Evaluate the following limits, where a and b are fixed real numbers. lim tS3 a a4t - 2 t - 3 b16 + t - t22b

3952. Other techniques Evaluate the following limits, where a and b are fixed real numbers. lim xSa x - a 1x - 1a , a 7 0

3952. Other techniques Evaluate the following limits, where a and b are fixed real numbers. lim xSa x2 - a2 1x - 1a , a 7 0

3952. Other techniques Evaluate the following limits, where a and b are fixed real numbers. lim hS0 116 + h - 4 h

3952. Other techniques Evaluate the following limits, where a and b are fixed real numbers. lim xSa x3 - a3 x - a

Slope of a tangent line a. Sketch a graph of y = 2x and carefully draw three secant lines connecting the points P10, 12 and Q1x, 2x2, for x = -3, -2, and -1. b. Find the slope of the line that passes through P10, 12 and Q1x, 2x2, for x _ 0. c. Complete the table and make a conjecture about the value of lim xS0- 2x - 1 x . x -1 -0.1 -0.01 -0.001 -0.0001 -0.00001 2x _ 1 x

Slope of a tangent line a. Sketch a graph of y = 3x and carefully draw four secant lines connecting the points P10, 12 and Q1x, 3x2, for x = -2, -1, 1, and 2. b. Find the slope of the line that passes through P10, 12 and Q1x, 3x2, for x _ 0. c. Complete the table and make a conjecture about the value of lim xS0 3x - 1 x . x -0.1 -0.01 -0.001 -0.0001 0.0001 0.001 0.01 0.1 3x _ 1 x

Applying the Squeeze Theorem a. Show that - _ x _ x sin 1 x _ x _, for x _ 0. b. Illustrate the inequalities in part (a) with a graph. c. Use the Squeeze Theorem to show that lim xS0 x sin 1 x = 0.

A cosine limit by the Squeeze Theorem It can be shown that 1 - x2 2 cos x 1, for x near 0. a. Illustrate these inequalities with a graph. b. Use these inequalities to evaluate lim xS0 cos x.

A sine limit by the Squeeze Theorem It can be shown that 1 - x2 6 sin x x 1, for x near 0. a. Illustrate these inequalities with a graph. b. Use these inequalities to evaluate lim xS0 sin x x .

A logarithm limit by the Squeeze Theorem a. Draw a graph to verify that - _ x _ x2 ln x2 _ x _, for -1 x 1, where x _ 0. b. Use the Squeeze Theorem to evaluate lim xS0 x2 ln x2.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume a and L are finite numbers. a. If lim xSa f 1x2 = L, then f 1a2 = L. b. If lim xSaf 1x2 = L, then lim xSa+ f 1x2 = L. c. If lim xSa f 1x2 = L and lim xSa g1x2 = L, then f 1a2 = g1a2. d. The limit lim xSa f 1x2 g1x2 does not exist if g1a2 = 0. e. If lim xS1+ 2f 1x2 = 4lim xS1+ f 1x2, it follows that lim xS1 2f 1x2 = 4lim xS1 f 1x2.

6067. Evaluating limits Evaluate the following limits, where c and k are constants. lim hS0 100 110h - 1211 + 2

6067. Evaluating limits Evaluate the following limits, where c and k are constants. lim xS2 15x - 623>2

6067. Evaluating limits Evaluate the following limits, where c and k are constants. lim xS3 1 x 2 + 2x - 1 15 x - 3

6067. Evaluating limits Evaluate the following limits, where c and k are constants. lim xS1 110x - 9 - 1 x - 1

6067. Evaluating limits Evaluate the following limits, where c and k are constants. lim xS2 a 1 x - 2 - 2 x2 - 2x b

6067. Evaluating limits Evaluate the following limits, where c and k are constants. lim hS0 15 + h22 - 25 h

6067. Evaluating limits Evaluate the following limits, where c and k are constants. lim xSc x2 - 2cx + c2 x - c

6067. Evaluating limits Evaluate the following limits, where c and k are constants. lim wS-k w2 + 5kw + 4k2 w2 + kw , for k _ 0

Finding a constant Suppose f 1x2 = b 3x + b if x 2 x - 2 if x 7 2. Determine a value of the constant b for which lim xS2 f 1x2 exists and state the value of the limit, if possible.

Finding a constant Suppose g1x2 = b x2 - 5x if x -1 ax3 - 7 if x 7 -1. Determine a value of the constant a for which lim xS-1 g1x2 exists and state the value of the limit, if possible.

7076. Useful factorization formula Calculate the following limits using the factorization formula xn - an = 1x - a21xn - 1 + xn - 2a + xn - 3a2 + g+ xan - 2 + an - 12, where n is a positive integer and a is a real number. lim xS2 x5 - 32 x - 2

7076. Useful factorization formula Calculate the following limits using the factorization formula xn - an = 1x - a21xn - 1 + xn - 2a + xn - 3a2 + g+ xan - 2 + an - 12, where n is a positive integer and a is a real number. lim xS1 x6 - 1 x - 1

7076. Useful factorization formula Calculate the following limits using the factorization formula xn - an = 1x - a21xn - 1 + xn - 2a + xn - 3a2 + g+ xan - 2 + an - 12, where n is a positive integer and a is a real number. lim xS-1 x7 + 1 x + 1 (Hint: Use the formula for x7 - a7 with a = -1.)

7076. Useful factorization formula Calculate the following limits using the factorization formula xn - an = 1x - a21xn - 1 + xn - 2a + xn - 3a2 + g+ xan - 2 + an - 12, where n is a positive integer and a is a real number. lim xSa x5 - a5 x - a

7076. Useful factorization formula Calculate the following limits using the factorization formula xn - an = 1x - a21xn - 1 + xn - 2a + xn - 3a2 + g+ xan - 2 + an - 12, where n is a positive integer and a is a real number. lim xSa xn - an x - a , for any positive integer n

7076. Useful factorization formula Calculate the following limits using the factorization formula xn - an = 1x - a21xn - 1 + xn - 2a + xn - 3a2 + g+ xan - 2 + an - 12, where n is a positive integer and a is a real number. lim xS1 23 x - 1 x - 1 (Hint: x - 1 = 123 x23-1123.)

7076. Useful factorization formula Calculate the following limits using the factorization formula xn - an = 1x - a21xn - 1 + xn - 2a + xn - 3a2 + g+ xan - 2 + an - 12, where n is a positive integer and a is a real number. lim xS16 24 x - 2 x - 16

7780. Limits involving conjugates Evaluate the following limits. lim xS1 x - 1 1x - 1

7780. Limits involving conjugates Evaluate the following limits. lim xS1 x - 1 14x + 5 - 3

7780. Limits involving conjugates Evaluate the following limits. lim xS4 31x - 421x + 5 3 - 1x + 5

7780. Limits involving conjugates Evaluate the following limits. lim xS0 x 1cx + 1 - 1 , where c is a nonzero constant

Creating functions satisfying given limit conditions Find functions f and g such that lim xS1 f 1x2 = 0 and lim xS1 1 f 1x2 g1x22 = 5.

Creating functions satisfying given limit conditions Find a function f satisfying lim xS1 a f 1x2 x - 1 b = 2.

Finding constants Find constants b and c in the polynomial p1x2 = x2 + bx + c such that lim xS2 p1x2 x - 2 = 6. Are the constants unique?

A problem from relativity theory Suppose a spaceship of length L0 travels at a high speed v relative to an observer. To the observer, the ship appears to have a smaller length given by the Lorentz contraction formula L = L0B 1 - v2 c2 , where c is the speed of light. a. What is the observed length L of the ship if it is traveling at 50% of the speed of light? b. What is the observed length L of the ship if it is traveling at 75% of the speed of light? c. In parts (a) and (b), what happens to L as the speed of the ship increases? d. Find lim vSc- L0B 1 - v2 c2 and explain the significance of this limit.

Limit of the radius of a cylinder A right circular cylinder with a height of 10 cm and a surface area of S cm2 has a radius given by r1S2 = 1 2 a A 100 + 2S p - 10b. Find lim SS0+ r1S2 and interpret your result.

Torricellis Law A cylindrical tank is filled with water to a depth of 9 meters. At t = 0, a drain in the bottom of the tank is opened and water flows out of the tank. The depth of water in the tank (measured from the bottom of the tank) t seconds after the drain is opened is approximated by d1t2 = 13 - 0.015t22, for 0 t 200. Evaluate and interpret lim tS200- d1t2.

Electric field The magnitude of the electric field at a point x meters from the midpoint of a 0.1-m line of charge is given by E1x2 = 4.35 x2x2 + 0.01 (in units of newtons per coulomb, N>C). Evaluate lim xS10 E1x2.

8889. Limits of composite functions If lim xS1 f 1x2 = 4, find lim xS-1 f 1x22.

8889. Limits of composite functions Suppose g1x2 = f 11 - x2, for all x, lim xS1+ f 1x2 = 4, and lim xS1- f 1x2 = 6. Find lim xS0+ g1x2 and lim xS0- g1x2.

Two trigonometric inequalities Consider the angle u in standard position in a unit circle, where 0 u 6 p>2 or -p>2 6 u 6 0 (use both figures). a. Show that _AC_ = _ sin u _, for -p>2 6 u 6 p>2. (Hint: Consider the cases 0 u 6 p>2 and -p>2 6 u 6 0 separately.) b. Show that _ sin u _ 6 _ u _, for -p>2 6 u 6 p>2. (Hint: The length of arc AB is u, if 0 u 6 p>2, and -u, if -p>2 6 u 6 0.) c. Conclude that - _ u _ sin u _ u _, for -p>2 6 u 6 p>2. d. Show that 0 1 - cos u _ u _, for -p>2 6 u 6 p>2.

Theorem 2.4a Given the polynomial p1x2 = bn xn + bn - 1xn - 1 + g+ b1x + b0, prove that lim xSa p1x2 = p1a2 for any value of a.