91Ó°ÊÓ

Estimating Limits Graphically Use a graphing device to determine whether the limit exists. If the limit exists, estimate its value to two decimal places. $$\lim _{x \rightarrow 3} \frac{|x-3|}{x-3}$$

Find the following for the given function \(f:\) (a) \(f^{\prime}(a),\) where \(a\) is in the domain of \(f,\) and (b) \(f^{\prime}(3)\) and \(f^{\prime}(4)\) $$f(x)=x^{2}+2 x$$

Finding Limits Evaluate the limit if it exists. $$\lim _{x \rightarrow 7} \frac{\sqrt{x+2}-3}{x-7}$$

Limits of Sequences If the sequence with the given \(n\) th term is convergent, find its limit. If it is divergent, explain why. $$a_{n}=\frac{(-1)^{n}}{n}$$

Find the following for the given function \(f:\) (a) \(f^{\prime}(a),\) where \(a\) is in the domain of \(f,\) and (b) \(f^{\prime}(3)\) and \(f^{\prime}(4)\) $$f(x)=-\frac{1}{x^{2}}$$

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

Finding Limits Evaluate the limit if it exists. $$\lim _{h \rightarrow 0} \frac{\sqrt{1+h}-1}{h}$$

Limits of Sequences If the sequence with the given \(n\) th term is convergent, find its limit. If it is divergent, explain why. $$a_{n}=\sin (n \pi / 2)$$

Finding Limits Evaluate the limit if it exists. $$\lim _{x \rightarrow-4} \frac{\frac{1}{4}+\frac{1}{x}}{4+x}$$

Find the following for the given function \(f:\) (a) \(f^{\prime}(a),\) where \(a\) is in the domain of \(f,\) and (b) \(f^{\prime}(3)\) and \(f^{\prime}(4)\) $$f(x)=\frac{x}{x+1}$$