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Chapter 2: Problem 8
Suppose \(\lim _{x \rightarrow 2} f(x)=\lim _{x \rightarrow 2} h(x)=5 .\) Find \(\lim _{x \rightarrow 2} g(x),\) where \(f(x) \leq g(x) \leq h(x),\) for all \(x\).
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Classify the discontinuities in the following functions at the given points. See Exercises \(91-92.\) $$h(x)=\frac{x^{3}-4 x^{2}+4 x}{x(x-1)} ; x=0 \text { and } x=1$$
We say that \(\lim _{x \rightarrow \infty} f(x)=\infty\) if for any positive number \(M,\) there is \(a\) corresponding \(N>0\) such that $$f(x)>M \quad \text { whenever } \quad x>N$$ Use this definition to prove the following statements. $$\lim _{x \rightarrow \infty} \frac{x}{100}=\infty$$
We say that \(\lim _{x \rightarrow \infty} f(x)=\infty\) if for any positive number \(M,\) there is \(a\) corresponding \(N>0\) such that $$f(x)>M \quad \text { whenever } \quad x>N$$ Use this definition to prove the following statements. $$\lim _{x \rightarrow \infty} \frac{x^{2}+x}{x}=\infty$$
Torricelli's Law A cylindrical tank is filled with water to a depth of 9 meters. At \(t=0,\) a drain in the bottom of the tank is opened and water flows out of the tank. The depth of water in the tank (measured from the bottom of the tank) \(t\) seconds after the drain is opened is approximated by \(d(t)=(3-0.015 t)^{2},\) for \(0 \leq t \leq 200\). Evaluate and interpret \(\lim _{t \rightarrow 200^{-}} d(t)\).
Asymptotes Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions. $$f(x)=\frac{x^{2}-3 x+2}{x^{10}-x^{9}}$$
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