91Ó°ÊÓ

Find the limit of the following sequences or state that the limit does not exist. $$\begin{aligned} &\left\\{0, \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \ldots\right\\}, \text { which is defined by } f(n)=\frac{n-1}{n}, \text { for }\\\ &n=1,2,3, \dots \end{aligned}$$

Evaluate the following limits or state that they do not exist. $$\lim _{x \rightarrow \infty} \frac{\tan ^{-1} x}{x}$$

Evaluate the following limits. \(\lim _{x \rightarrow 1} \frac{x-1}{\sqrt{x}-1}\)

Find the limit of the following sequences or state that the limit does not exist. $$\begin{aligned} &\left\\{\frac{1}{2}, \frac{4}{3}, \frac{9}{4}, \frac{16}{5}, \ldots\right\\}, \text { which is defined by } f(n)=\frac{n^{2}}{n+1}, \text { for }\\\ &n=1,2,3, \dots \end{aligned}$$

Evaluate the following limits. \(\lim _{x \rightarrow 1} \frac{x-1}{\sqrt{4 x+5}-3}\)

Evaluate the following limits or state that they do not exist. $$\lim _{t \rightarrow \infty} \frac{\cos t}{e^{3 t}}$$

Evaluate the following limits. \(\lim _{x \rightarrow 4} \frac{3(x-4) \sqrt{x+5}}{3-\sqrt{x+5}}\)

Evaluate the following limits or state that they do not exist. $$\lim _{x \rightarrow 1^{-}} \frac{x}{\ln x}$$

Find the limit of the following sequences or state that the limit does not exist. $$\begin{aligned} &\left\\{2, \frac{3}{4}, \frac{4}{9}, \frac{5}{16}, \ldots\right\\}, \text { which is defined by } f(n)=\frac{n+1}{n^{2}}, \text { for }\\\ &n=1,2,3, \ldots \end{aligned}$$

Evaluate the following limits. \(\lim _{x \rightarrow 0} \frac{x}{\sqrt{c x+1}-1},\) where \(c\) is a nonzero constant