91Ó°ÊÓ

Evaluate the limit, if it exists. $$\lim _{t \rightarrow 0}\left(\frac{1}{t}-\frac{1}{t^{2}+t}\right)$$

If the sequence is convergent, find its limit. If it is divergent, explain why. $$a_{n}=\frac{n^{2}}{n+1}$$

Find the limit and use a graphing device to confirm your result graphically. $$\lim _{x \rightarrow 1} \frac{x^{2}-1}{\sqrt{x}-1}$$

Find \(f^{\prime}(a),\) where \(a\) is in the domain of \(f .\) $$f(x)=\frac{x}{x+1}$$

Use a graphing device to determine whether the limit exists. If the limit exists, estimate its value to two decimal places. $$\lim _{x \rightarrow 0} \cos \frac{1}{x}$$

Use a graphing device to determine whether the limit exists. If the limit exists, estimate its value to two decimal places. $$\lim _{x \rightarrow 0} \frac{1}{1+e^{1 / x}}$$

Find the limit and use a graphing device to confirm your result graphically. $$\lim _{x \rightarrow 0} \frac{(4+x)^{3}-64}{x}$$

If the sequence is convergent, find its limit. If it is divergent, explain why. $$a_{n}=\frac{n-1}{n^{3}+1}$$

If the sequence is convergent, find its limit. If it is divergent, explain why. $$a_{n}=\frac{1}{3^{n}}$$

Graph the piecewise-defined function and use your graph to find the values of the limits, if they exist. $$f(x)=\left\\{\begin{array}{ll} x^{2} & \text { if } x \leq 2 \\ 6-x & \text { if } x>2 \end{array}\right.$$ (a) \(\lim _{x \rightarrow 2^{-}} f(x)\) (b) \(\lim _{x \rightarrow 2^{+}} f(x)\) (c) \(\lim _{x \rightarrow 2} f(x)\)