91Ó°ÊÓ

Decide which of the given one-sided or twosided limits exist as numbers, which as \(\infty\), which as \(-\infty\), and which do not exist. Where the limit is a number, evaluate it. $$ \lim _{x \rightarrow 5^{+}} \frac{1}{x \sqrt{x-5}} $$

Evaluate the function at \(0.1,0.01\), and \(0.001\), and at \(-0.1,-0.01\), and \(-0.001\). Then guess the value of \(\lim _{x \rightarrow 0} f(x) .\) \(f(x)=\frac{1-\cos x}{x}\)

Use the results of this section to evaluate the limit. $$ \lim _{x \rightarrow 0} \frac{\sin x}{\sin 2 x} $$

Evaluate the function at \(0.1,0.01\), and \(0.001\), and at \(-0.1,-0.01\), and \(-0.001\). Then guess the value of \(\lim _{x \rightarrow 0} f(x) .\) \(f(x)=\frac{1-\cos x}{x^{2}}\)

Explain why \(f\) is continuous on the given interval or intervals $$ f(x)=\sqrt{16 x^{4}-x^{2}} ;\left(-\infty,-\frac{1}{4}\right],\left[\frac{1}{4}, \infty\right) $$

In Exercises \(f(t)\) is the position at time \(t\) of an object moving along the \(x\) axis. Find the velocity of the object at time \(t_{0}\). $$ f(t)=-3 t+\frac{1}{5} ; t_{0}=0 $$

Decide which of the given one-sided or twosided limits exist as numbers, which as \(\infty\), which as \(-\infty\), and which do not exist. Where the limit is a number, evaluate it. $$ \lim _{x \rightarrow 1^{+}}\left(\sqrt{x^{2}-x}+x\right) $$

In Exercises \(f(t)\) is the position at time \(t\) of an object moving along the \(x\) axis. Find the velocity of the object at time \(t_{0}\). $$ f(t)=-16 t^{2} ; t_{0}=4 $$

Decide which of the given one-sided or twosided limits exist as numbers, which as \(\infty\), which as \(-\infty\), and which do not exist. Where the limit is a number, evaluate it. $$ \lim _{x \rightarrow 0^{+}} \sqrt{x \cos 2 x} $$

Evaluate the function at \(0.1,0.01\), and \(0.001\), and at \(-0.1,-0.01\), and \(-0.001\). Then guess the value of \(\lim _{x \rightarrow 0} f(x) .\) \(f(x)=\frac{\sin 3 x}{\sin 5 x}\)