91Ó°ÊÓ

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

$\underset{x→1.7}{\lim }\left(3.1x^2-4x+0.8\right)$.

Consider the graph of the function:

Find the limits of function.

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.

${∞}^{∞}$

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| < $\mathrm{Î\mu }$.

$\underset{x→1}{\lim }(3x+8)=11$

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

f is continuous on [0, 2) but not on [0, 2].

Find the limit by hand.

$\underset{x→0}{\lim }x\sin \frac{1}{x}$

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.

role="math" localid="1648286746185" $\underset{x→-∞}{\lim }\frac{1}{x^2+1}=0$

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.

$f\left(x\right)={\tan }^{-1}\left(3x\right)+1$

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| < $\mathrm{Î\mu }$.

$\underset{x→0}{\lim }(5x^2-1)=-1$

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.

$\underset{x→∞}{\lim }\left(x^3+x+1\right)=∞$