91Ó°ÊÓ

Calculate each limit in Exercises 35-80.

$\underset{x→∞}{\lim }\left(\sqrt{x}-x\right)$

For each limit statement , use algebra to find δ > 0 in terms of $\mathrm{Î\mu }$> 0 so that if 0 < |x − c| < δ, then | f(x) − L| < $\mathrm{Î\mu }$.

$\underset{x→3}{\lim }(x^2-2x-3)=0;\mathrm{you}\mathrm{may}\mathrm{assume}\mathrm{δ}≤1$

Indeterminate forms: Identify which of the limit forms listed here are indeterminate. For each form that is not indeterminate, describe the behavior of a limit of that form.

$\frac{1}{∞}.$

Calculate each of the limits in Exercises $29-70$.

$\underset{x→\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\lim }\frac{4x-2}{6x^2+x-2}$.

Find the limit of the expression using table

$\underset{x->2^{-}}{\lim }(x^2+x+1)$

Write each limit in Exercises 19–42 as a formal statement involving $\mathrm{δ},\mathrm{Î\mu },N,\mathrm{and}/\mathrm{or}M,$and sketch a graph that illustrates the roles of these constants. In the last six exercises you may take function to be any appropriate graph.

$\underset{x→-∞}{\lim }f\left(x\right)=L$

If $\underset{x→0^{+}}{\lim }f\left(x\right)=-2$, $\underset{x→0^{-}}{\lim }f\left(x\right)=3$, and $f\left(0\right)=-2$, what can you say about $\underset{x→0}{\lim }f\left(x\right)$?

Suppose you show that $\left|\right(1-2x)-(-5\left)\right|<0.05$ for all x with $0<|x-3|<0.025$Explain why this does not prove thatrole="math" localid="1648055981096" $\underset{x→3}{lim}(1-2x)=-5$

Describe the punctured interval around $x=2$that has a radius of $3$ and the punctured interval around$x=4$ that has a radius of $0.25$.

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$M≠0$