91Ó°ÊÓ

Explain the meaning of \(\lim _{-0^{-}} f(x)=L\).

Assume \(\lim _{x \rightarrow 1} f(x)=8, \lim _{x \rightarrow 1} g(x)=3,\) and \(\lim _{x \rightarrow 1} h(x)=2 .\) Compute the following limits and state the limit laws used to justify your computations. $$\lim _{x \rightarrow 1} \sqrt[3]{f(x) g(x)+3}$$

Determine the following limits at infinity. $$\lim _{t \rightarrow \infty} e^{t}, \lim _{t \rightarrow-\infty} e^{t}, \text { and } \lim _{t \rightarrow \infty} e^{-t}$$

If \(\lim _{x \rightarrow a^{-}} f(x)=L\) and \(\lim _{x \rightarrow a^{+}} f(x)=M,\) where \(L\) and \(M\) are finite real numbers, then how are \(L\) and \(M\) related if \(\lim f(x)\) exists?

Assume \(\lim _{x \rightarrow 1} f(x)=8, \lim _{x \rightarrow 1} g(x)=3,\) and \(\lim _{x \rightarrow 1} h(x)=2 .\) Compute the following limits and state the limit laws used to justify your computations. $$\lim _{x \rightarrow 1}(f(x))^{2 / 3}$$

Finding \(\delta\) for a given \(\varepsilon\) using a graph Let \(f(x)=x^{3}+3\) and note that \(\lim _{x \rightarrow 0} f(x)=3 .\) For each value of \(\varepsilon,\) use a graphing utility to find all values of \(\delta>0\) such that \(|f(x)-3|<\varepsilon\) whenever \(0<|x-0|<\delta .\) Sketch graphs illustrating your work. a. \(\varepsilon=1\) b. \(\varepsilon=0.5\)

Average velocity The position of an object moving vertically along a line is given by the function \(s(t)=-16 t^{2}+128 t\). Find the average velocity of the object over the following intervals. a. [1,4] b. [1,3] c. [1,2] d. \([1,1+h],\) where \(h>0\) is a real number

Finding \(\delta\) for a given \(\varepsilon\) using a graph Let \(g(x)=2 x^{3}-12 x^{2}+26 x+4\) and note that \(\lim _{x \rightarrow 2} g(x)=24\) For each value of \(\varepsilon\), use a graphing utility to find all values of \(\delta>0\) such that \(|g(x)-24|<\varepsilon\) whenever \(0<|x-2|<\delta\) Sketch graphs illustrating your work. a. \(\varepsilon=1\) b. \(\varepsilon=0.5\)

How are \(\lim _{x \rightarrow a} p(x)\) and \(\lim _{x \rightarrow a^{+}} p(x)\) calculated if \(p\) is a polynomial function?

Average velocity The position of an object moving vertically along a line is given by the function \(s(t)=-4.9 t^{2}+30 t+20\) Find the average velocity of the object over the following intervals. a. [0,3] b. [0,2] c. [0,1] d. \([0, h],\) where \(h>0\) is a real number