 14.2.1: The limit of the product of two functions equals the of their limits
 14.2.2: lim xSc b = .
 14.2.3: lim xSc x = .
 14.2.4: True or False The limit of a polynomial function as x approaches 5 ...
 14.2.5: True or False The limit of a rational function at 5 equals the valu...
 14.2.6: True or False The limit of a quotient equals the quotient of the li...
 14.2.7: In 7 42, find each limit algebraically.
 14.2.8: In 7 42, find each limit algebraically.
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 14.2.39: In 7 42, find each limit algebraically.
 14.2.40: In 7 42, find each limit algebraically.
 14.2.41: In 7 42, find each limit algebraically.
 14.2.42: In 7 42, find each limit algebraically.
 14.2.43: In 4352, find the limit as x approaches c of the average rate of ch...
 14.2.44: In 4352, find the limit as x approaches c of the average rate of ch...
 14.2.45: In 4352, find the limit as x approaches c of the average rate of ch...
 14.2.46: In 4352, find the limit as x approaches c of the average rate of ch...
 14.2.47: In 4352, find the limit as x approaches c of the average rate of ch...
 14.2.48: In 4352, find the limit as x approaches c of the average rate of ch...
 14.2.49: In 4352, find the limit as x approaches c of the average rate of ch...
 14.2.50: In 4352, find the limit as x approaches c of the average rate of ch...
 14.2.51: In 4352, find the limit as x approaches c of the average rate of ch...
 14.2.52: In 4352, find the limit as x approaches c of the average rate of ch...
 14.2.53: In 5356, use the properties of limits and the facts that lim xS0 si...
 14.2.54: In 5356, use the properties of limits and the facts that lim xS0 si...
 14.2.55: In 5356, use the properties of limits and the facts that lim xS0 si...
 14.2.56: In 5356, use the properties of limits and the facts that lim xS0 si...
Solutions for Chapter 14.2: Algebra Techniques for Finding Limits
Full solutions for Precalculus Enhanced with Graphing Utilities  6th Edition
ISBN: 9780132854351
Solutions for Chapter 14.2: Algebra Techniques for Finding Limits
Get Full SolutionsThis textbook survival guide was created for the textbook: Precalculus Enhanced with Graphing Utilities, edition: 6. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 14.2: Algebra Techniques for Finding Limits includes 56 full stepbystep solutions. Since 56 problems in chapter 14.2: Algebra Techniques for Finding Limits have been answered, more than 53976 students have viewed full stepbystep solutions from this chapter. Precalculus Enhanced with Graphing Utilities was written by and is associated to the ISBN: 9780132854351.

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

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib IIĀ· Condition numbers measure the sensitivity of the output to change in the input.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

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

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.