91Ó°ÊÓ

Evaluate the following limits, where a and \(b\) are fixed real numbers. \(\lim _{x \rightarrow a} \frac{x-a}{\sqrt{x}-\sqrt{a}}, a>0\)

Evaluate each limit and justify your answer. $$\lim _{x \rightarrow 3} \sqrt{x^{2}+7}$$

Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. $$f(x)=\sin x$$

Use analytical methods and/or a graphing utility en identify the vertical asymptotes (if any) of the following functions. $$g(\theta)=\tan \frac{\pi \theta}{10}$$

Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. $$f(x)=\frac{50}{e^{2 x}}$$

Evaluate the following limits, where a and \(b\) are fixed real numbers. \(\lim _{x \rightarrow a} \frac{x^{2}-a^{2}}{\sqrt{x}-\sqrt{a}}, a>0\)

The limit at infinity \(\lim _{x \rightarrow \infty} f(x)=L\) means that for any \(\varepsilon>0\) there exists \(N>0\) such that $$|f(x)-L|<\varepsilon \quad \text { whenever } \quad x>N.$$ Use this definition to prove the following statements. $$\lim _{x \rightarrow \infty} \frac{10}{x}=0$$

Use analytical methods and/or a graphing utility en identify the vertical asymptotes (if any) of the following functions. $$q(s)=\frac{\pi}{s-\sin s}$$

Evaluate each limit and justify your answer. $$\lim _{t \rightarrow 2} \frac{t^{2}+5}{1+\sqrt{t^{2}+5}}$$

The limit at infinity \(\lim _{x \rightarrow \infty} f(x)=L\) means that for any \(\varepsilon>0\) there exists \(N>0\) such that $$|f(x)-L|<\varepsilon \quad \text { whenever } \quad x>N.$$ Use this definition to prove the following statements. $$\lim _{x \rightarrow \infty} \frac{2 x+1}{x}=2$$