91Ó°ÊÓ

Limits of basic functions: Fill in the blanks to complete the limit rules that follow. You may assume that k is positive .

$\underset{x→∞}{\lim }{x}^{-k}=?.$

In Exercises 3–6, $\underset{x→c}{\lim }f\left(x\right)=L$and $\underset{x→c}{\lim }g\left(x\right)=M$for some

real numbers L and M. What, if anything, can you say about$\underset{x→c}{\lim }\frac{f\left(x\right)}{g\left(x\right)}$in each case?

$L=0$and$M=0$.

Use the definition of the derivative to calculate the derivative of $f\left(x\right)=x^{-1/2}$at $c=4$. As in the previous calculation, you will need to multiply numerator and denominator by a conjugate at some point.

Fill in the blanks to complete each of the following theorem statements:

The Intermediate Value Theorem: If f is on a closed interval$[a,b]$, then for any K strictly between and , there exists at least one $c∈(a,b)$such that .

Find functions f and g and a real number c such that $\underset{x→c}{\lim }f\left(x\right)+\underset{x→c}{\lim }g\left(x\right)≠\underset{x→c}{\lim }\left(f\right(x)+g(x\left)\right)$. Does this example contradict the sum rule for limits? Why or why not?

what it means, in terms of limits, for a function f to have a vertical asymptote at x = c or a horizontal asymptote at y = L

Calculating limits: Find each limit by hand.

$\underset{x→-∞}{\lim }\frac{x^3+2x+1}{1-x^4}.$

Show that as $n→∞$we would expect the preceding expansion to approach

$1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\frac{1}{5!}+⋯.$

Find punctured intervals on which the function $f\left(x\right)=\frac{1}{x\ln \left(x+2\right)}$is defined, centered around

$\left(\mathrm{a}\right)x=0\left(\mathrm{b}\right)x=-1\left(\mathrm{c}\right)x=-1.5$

Why do we have $0<|x-c|<\mathrm{δ}$ instead of just $|x-c|<\mathrm{δ}$ in Definition 1.10?