91Ó°ÊÓ

Use tables of values to make educated guesses for each of the limits in Exercises 39–52.

$\underset{x→3^{+}}{\lim }\left(1-3x+x^2\right)$

In Exercises 39–44, use Theorem 1.16 and left and right limits to determine whether each function f is continuous at its break point(s). For each discontinuity of f, describe the type of discontinuity and any one-sided discontinuity.

$f\left(x\right)=\left\{\begin{array}{l}(x-3),ifx<0\\ -(x-3),ifxâ\copyright ¾0\end{array}.\right.$

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.

$\frac{∞}{1}.$

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| < $\mathrm{Î\mu }$.

$\underset{x→1}{\lim }2x^4=2;\mathrm{you}\mathrm{may}\mathrm{assume}\mathrm{δ}≤1$.

Calculate each limit in Exercises 35-80.

$\underset{x→∞}{\lim }\left(x^4-x^5\right)$

For each limit statement, use algebra to find δ or N in terms of $\mathrm{Î\mu }$or M, according to the appropriate formal limit definition.

$\underset{x→−2^{+}}{lim} (1+\sqrt{x+2})=1$ find δ in terms of $\mathrm{Î\mu }$.

In Exercises 39–44, use Theorem 1.16 and left and right limits to determine whether each function f is continuous at its break point(s). For each discontinuity of f, describe the type of discontinuity and any one-sided discontinuity.

$f\left(x\right)=\left\{\begin{array}{c}{x}^2,ifx<2\\ 4,ifx=2\\ 2x+1,ifx>2\end{array}.\right.$

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.

$0×∞$.

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

$\underset{x→1^{+}}{\lim }\frac{x-1}{\sqrt{x-1}}$.

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.

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