 2.4.1: Given that lim x l2 fsxd 4 lim x l2 tsxd 22 lim x l2 hsxd 0 find th...
 2.4.2: The graphs of f and t are given. Use them to evaluate each limit, i...
 2.4.3: 37 Evaluate the limit and justify each step by indicating the appro...
 2.4.4: 37 Evaluate the limit and justify each step by indicating the appro...
 2.4.5: 37 Evaluate the limit and justify each step by indicating the appro...
 2.4.6: 37 Evaluate the limit and justify each step by indicating the appro...
 2.4.7: 37 Evaluate the limit and justify each step by indicating the appro...
 2.4.8: (a) What is wrong with the following equation? x 2 1 x 2 6 x 2 2 x ...
 2.4.9: 924 Evaluate the limit, if it exists. 9. lim xl 5 x 2 2 6x 1 5 x 2 ...
 2.4.10: 924 Evaluate the limit, if it exists. 9. lim xl 5 x 2 2 6x 1 5 x 2 ...
 2.4.11: 924 Evaluate the limit, if it exists. 9. lim xl 5 x 2 2 6x 1 5 x 2 ...
 2.4.12: 924 Evaluate the limit, if it exists. 9. lim xl 5 x 2 2 6x 1 5 x 2 ...
 2.4.13: 924 Evaluate the limit, if it exists. 9. lim xl 5 x 2 2 6x 1 5 x 2 ...
 2.4.14: 924 Evaluate the limit, if it exists. 9. lim xl 5 x 2 2 6x 1 5 x 2 ...
 2.4.15: 924 Evaluate the limit, if it exists. 9. lim xl 5 x 2 2 6x 1 5 x 2 ...
 2.4.16: 924 Evaluate the limit, if it exists. 9. lim xl 5 x 2 2 6x 1 5 x 2 ...
 2.4.17: 924 Evaluate the limit, if it exists. 9. lim xl 5 x 2 2 6x 1 5 x 2 ...
 2.4.18: 924 Evaluate the limit, if it exists. 9. lim xl 5 x 2 2 6x 1 5 x 2 ...
 2.4.19: 924 Evaluate the limit, if it exists. 9. lim xl 5 x 2 2 6x 1 5 x 2 ...
 2.4.20: 924 Evaluate the limit, if it exists. 9. lim xl 5 x 2 2 6x 1 5 x 2 ...
 2.4.21: 924 Evaluate the limit, if it exists. 9. lim xl 5 x 2 2 6x 1 5 x 2 ...
 2.4.22: 924 Evaluate the limit, if it exists. 9. lim xl 5 x 2 2 6x 1 5 x 2 ...
 2.4.23: 924 Evaluate the limit, if it exists. 9. lim xl 5 x 2 2 6x 1 5 x 2 ...
 2.4.24: 924 Evaluate the limit, if it exists. 9. lim xl 5 x 2 2 6x 1 5 x 2 ...
 2.4.25: (a) Estimate the value of lim xl 0 x s1 1 3x 2 1 by graphing the fu...
 2.4.26: (a) Use a graph of fsxd s3 1 x 2 s3 x to estimate the value of limx...
 2.4.27: Use the Squeeze Theorem to show that limxl0 sx 2 cos 20xd 0. Illust...
 2.4.28: Use the Squeeze Theorem to show that lim xl 0 sx 3 1 x 2 sin x 0 Il...
 2.4.29: If 4x 2 9 < fsxd < x 2 2 4x 1 7 for x > 0, find lim xl 4 fsxd.
 2.4.30: If 2x < tsxd < x 4 2 x 2 1 2 for all x, evaluate lim xl1 tsxd.
 2.4.31: Prove that limxl 0 x 4 cos 2 x 0.
 2.4.32: Gene regulation Genes produce molecules called mRNA that go on to p...
 2.4.33: 3336 Find the limit, if it exists. If the limit does not exist, exp...
 2.4.34: 3336 Find the limit, if it exists. If the limit does not exist, exp...
 2.4.35: 3336 Find the limit, if it exists. If the limit does not exist, exp...
 2.4.36: 3336 Find the limit, if it exists. If the limit does not exist, exp...
 2.4.37: Let tsxd x 2 1 x 2 6  x 2 2  . (a) Find (i) lim xl 21 tsxd (ii) l...
 2.4.38: Let fsxd H x 2 1 1 sx 2 2d 2 if x , 1 if x > 1 (a) Find limx l12 fs...
 2.4.39: 3944 Find the limit. 39. lim xl0 sin 3x x 40. lim xl0 sin 4x sin 6x...
 2.4.40: 3944 Find the limit. 39. lim xl0 sin 3x x 40. lim xl0 sin 4x sin 6x...
 2.4.41: 3944 Find the limit. 39. lim xl0 sin 3x x 40. lim xl0 sin 4x sin 6x...
 2.4.42: 3944 Find the limit. 39. lim xl0 sin 3x x 40. lim xl0 sin 4x sin 6x...
 2.4.43: 3944 Find the limit. 39. lim xl0 sin 3x x 40. lim xl0 sin 4x sin 6x...
 2.4.44: 3944 Find the limit. 39. lim xl0 sin 3x x 40. lim xl0 sin 4x sin 6x...
 2.4.45: (a) If p is a polynomial, show that limxla psxd psad. (b) If r is a...
 2.4.46: In the theory of relativity, the Lorentz contraction formula L L0s1...
 2.4.47: To prove that sine has the Direct Substitution Property we need to ...
 2.4.48: Prove that cosine has the Direct Substitution Property.
 2.4.49: If lim xl1 fsxd 2 8 x 2 1 10, find lim xl1 fsx
 2.4.50: If limxl 0 fsxd x 2 5, find the following limits. (a) lim x l0 fsxd...
 2.4.51: Is there a number a such that lim xl22 3x 2 1 ax 1 a 1 3 x 2 1 x 2 ...
 2.4.52: The figure shows a fixed circle C1 with equation sx 2 1d 2 1 y 2 1 ...
Solutions for Chapter 2.4: Limits: Algebraic Methods
Full solutions for Biocalculus: Calculus for Life Sciences  1st Edition
ISBN: 9781133109631
Solutions for Chapter 2.4: Limits: Algebraic Methods
Get Full SolutionsBiocalculus: Calculus for Life Sciences was written by and is associated to the ISBN: 9781133109631. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 2.4: Limits: Algebraic Methods includes 52 full stepbystep solutions. Since 52 problems in chapter 2.4: Limits: Algebraic Methods have been answered, more than 26132 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Biocalculus: Calculus for Life Sciences , edition: 1.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B IIĀ·

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.