5354 These exercises develop some versions of the substitution principle, a useful tool

Chapter 1, Problem 54

(choose chapter or problem)

Focus on Concepts

These exercises develop some versions of the substitution principle, a useful tool for the evaluation of limits.

 (a) Explain why we can evaluate \(\lim _{x \rightarrow+\infty} e^{x^{2}}\)  by making the substitution \(t=-x^{2}\) and writing

\(\lim _{x \rightarrow+\infty} e^{-x^{2}}=\lim _{t \rightarrow-\infty} e^{t}=0\)

(b) Suppose \(g(x) \rightarrow-\infty\) as \(g(x) \rightarrow+\infty\). Given any function \(f(x)\), explain why we can evaluate \(\lim _{x \rightarrow+\infty} f[g(x)]\) by substituting \(t=g(x)\) and writing

\(\lim _{x \rightarrow+\infty} f[g(x)]=\lim _{t \rightarrow-\infty} f(t)\)

(Here, "equality" is interpreted to mean that either both limits exist and are equal or that both limits fail to exist.)
(c) Why does the result in part (b) remain valid if \(\lim _{x \rightarrow+\infty}\) is replaced everywhere by one of \(\lim _{x \rightarrow-\infty}, \lim _{x \rightarrow c}, \lim _{x \rightarrow c^{-}}\), or \(\lim _{x \rightarrow c^{+}}\)?

Equation Transcription:

 

Text Transcription:

lim_x rightarrow + infinity e^x^2

t = -x^2

lim_x rightarrow + infinity e^-x^2 = lim_t rightarrow + infinity e^t = 0

g(x) rightarrow - infinity

x rightarrow + infinity

f(x)

lim_x rightarrow + infinity f[g(x)]

t = g(x)

lim_x rightarrow + infinity f[g(x)] = lim_t rightarrow - infinity f(t)

lim_x rightarrow + infinity

lim_x rightarrow -infinity, lim_x rightarrow c, lim_x rightarrow c^-

lim_x rightarrow c^+