91Ó°ÊÓ

Find the limits. $$ \lim _{h \rightarrow 0} \frac{\sin h}{2 h} $$

Sketch a possible graph for a function \(f\) with the specified properties. (Many different solutions are possible.) (i) the domain of \(f\) is \((0,+\infty)\) (ii) \(f(1)=0\) (iii) the \(y\) -axis is a vertical asymptote for the graph of \(f\) (iv) \(f(x)<0\) if \(0

Find the limits. $$ \lim _{\theta \rightarrow 0^{+}} \frac{\sin \theta}{\theta^{2}} $$

Determine whether the statement is true or false. Explain your answer. If \(f\) and \(g\) are discontinuous at \(x=c\), then so is \(f+g\).

Find the limits. $$ \lim _{x \rightarrow 4^{-}} \frac{3-x}{x^{2}-2 x-8} $$

Sketch a possible graph for a function \(f\) with the specified properties. (Many different solutions are possible.) (i) \(f(-3)=f(0)=f(2)=0\) (ii) \(\lim _{x \rightarrow-2^{-}} f(x)=+\infty\) and \(\lim _{x \rightarrow-2^{+}} f(x)=-\infty\) (iii) \(\lim _{x \rightarrow 1} f(x)=+\infty\)

Find the limits. $$ \lim _{x \rightarrow-\infty} \frac{\sqrt{5 x^{2}-2}}{x+3} $$

Sketch a possible graph for a function \(f\) with the specified properties. (Many different solutions are possible.) (i) \(f(-1)=0, f(0)=1, f(1)=0\) (ii) \(\lim _{x \rightarrow-1^{-}} f(x)=0\) and \(\lim _{x \rightarrow-1^{+}} f(x)=+\infty\) (iii) \(\lim _{x \rightarrow 1^{-}} f(x)=1\) and \(\lim _{x \rightarrow 1^{+}} f(x)=+\infty\)

Find the limits. $$ \lim _{x \rightarrow+\infty} \frac{\sqrt{5 x^{2}-2}}{x+3} $$

Find the limits. $$ \lim _{\theta \rightarrow 0} \frac{\sin ^{2} \theta}{\theta} $$