91Ó°ÊÓ

Find the limit by hand.

$\underset{x→0}{\lim }\frac{x^2}{e^x-1}$

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.

$\frac{0}{1}$

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.

role="math" localid="1648286577419" $\underset{x→0^{-}}{\lim }\frac{1}{x}=-∞$

Consider the limit values below:

$\underset{x→-∞}{\lim }f\left(x\right)=-2,\underset{x→3}{\lim }f\left(x\right)=∞\text{and}f\left(2\right)=-1$

The strategy is to sketch the graph of the function having the above limit values

Sketch the graph of a function f described in Exercises 27–32, if possible. If it is not possible, explain why not.

f is left continuous at x = 2 but not continuous at x = 2, and f(2) = 3.

Find the roots, discontinuities, and horizontal and vertical asymptotes of the functions in Exercises 23–34. Support your answers by explicitly computing any relevant limits.

$f\left(x\right)=\frac{2}{4+e^{-2x}}$

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

$\underset{x→0}{\lim }\left(x^2-1\right)$.

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| < $\mathrm{Î\mu }$.

$\underset{x→3}{\lim }(x+5)=8$

Find the limit by hand.

$\underset{x→∞}{\lim }\sin x$

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.

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