91Ó°ÊÓ

Prove the given limit using an \(\varepsilon-\delta\) proof. $$ \lim _{x \rightarrow 3}\left(x^{2}-3\right)=6 $$

Use the following information to evaluate the given limit, when possible. If it is not possible to determine the limit, state why not. $$ \begin{array}{ll} *\lim _{x \rightarrow 9} f(x)=6, & \lim _{x \rightarrow 6} f(x)=9, \quad f(9)=6 \\ *\lim _{x \rightarrow 9} g(x)=3, & \lim _{x \rightarrow 6} g(x)=3, \quad g(6)=9 \end{array} $$ $$ \lim _{x \rightarrow 9}(f(x)+g(x)) $$

Approximate the given limits both numerically and graphically. $$ \lim _{x \rightarrow 0} x^{3}-3 x^{2}+x-5 $$

Prove the given limit using an \(\varepsilon-\delta\) proof. $$ \lim _{x \rightarrow 4}\left(x^{2}+x-5\right)=15 $$

Use the following information to evaluate the given limit, when possible. If it is not possible to determine the limit, state why not. $$ \begin{array}{ll} *\lim _{x \rightarrow 9} f(x)=6, & \lim _{x \rightarrow 6} f(x)=9, \quad f(9)=6 \\ *\lim _{x \rightarrow 9} g(x)=3, & \lim _{x \rightarrow 6} g(x)=3, \quad g(6)=9 \end{array} $$ $$ \lim _{x \rightarrow 9}(3 f(x) / g(x)) $$

Let \(\lim _{x \rightarrow 7} f(x)=\infty\). Explain how we know that \(f\) is/is not continuous at \(x=7\).

Approximate the given limits both numerically and graphically. $$ \lim _{x \rightarrow 0} \frac{x+1}{x^{2}+3 x} $$

Use the following information to evaluate the given limit, when possible. If it is not possible to determine the limit, state why not. $$ \begin{array}{ll} *\lim _{x \rightarrow 9} f(x)=6, & \lim _{x \rightarrow 6} f(x)=9, \quad f(9)=6 \\ *\lim _{x \rightarrow 9} g(x)=3, & \lim _{x \rightarrow 6} g(x)=3, \quad g(6)=9 \end{array} $$ $$ \lim _{x \rightarrow 9}\left(\frac{f(x)-2 g(x)}{g(x)}\right) $$

Prove the given limit using an \(\varepsilon-\delta\) proof. $$ \lim _{x \rightarrow 1}\left(2 x^{2}+3 x+1\right)=6 $$

\(\mathrm{T} / \mathrm{F}:\) The sum of continuous functions is also continuous.