극한의 성질과 계산

In the following exercises, use direct substitution to show that each limit leads to the indeterminate form $0/0.$ Then, evaluate the limit.

$\lim_{x\to 4}\frac{{x}^{2}-16}{x-4}$

In the following exercises, use direct substitution to obtain an undefined expression. Then, use the method of (see text) to simplify the function to help determine the limit.

$\lim_{x\to {1}^{-}}\frac{2{x}^{2}+7x-4}{{x}^{2}+x-2}$

In the following exercises, use direct substitution to evaluate each limit.

$\lim_{x\to 7}{x}^{2}$

In the following exercises, use direct substitution to evaluate each limit.

$\lim_{x\to -2}(4{x}^{2}-1)$

In the following exercises, use direct substitution to evaluate each limit.

$\lim_{x\to 0}\frac{1}{1+\sin x}$

In the following exercises, use direct substitution to evaluate each limit.

$\lim_{x\to 2}{e}^{2x-{x}^{2}}$

In the following exercises, use direct substitution to evaluate each limit.

$\lim_{x\to 1}\frac{2-7x}{x+6}$

In the following exercises, use direct substitution to evaluate each limit.

$\lim_{x\to 3}\ln {e}^{3x}$

In the following exercises, use direct substitution to obtain an undefined expression. Then, use the method of (see text) to simplify the function to help determine the limit.

$\lim_{x\to {-2}^{-}}\frac{2{x}^{2}+7x-4}{{x}^{2}+x-2}$

In the following exercises, use direct substitution to obtain an undefined expression. Then, use the method of (see text) to simplify the function to help determine the limit.

$\lim_{x\to {-2}^{+}}\frac{2{x}^{2}+7x-4}{{x}^{2}+x-2}$

In the following exercises, use direct substitution to obtain an undefined expression. Then, use the method of (see text) to simplify the function to help determine the limit.

$\lim_{x\to {1}^{+}}\frac{2{x}^{2}+7x-4}{{x}^{2}+x-2}$

In the following exercises, assume that $\lim_{x\to 6}f(x)=4,\lim_{x\to 6}g(x)=9,$ and $\lim_{x\to 6}h(x)=6.$ Use these three facts and the limit laws to evaluate each limit.

$\lim_{x\to 6}2f(x)g(x)$

In the following exercises, assume that $\lim_{x\to 6}f(x)=4,\lim_{x\to 6}g(x)=9,$ and $\lim_{x\to 6}h(x)=6.$ Use these three facts and the limit laws to evaluate each limit.

$\lim_{x\to 6}\frac{g(x)-1}{f(x)}$

In the following exercises, assume that $\lim_{x\to 6}f(x)=4,\lim_{x\to 6}g(x)=9,$ and $\lim_{x\to 6}h(x)=6.$ Use these three facts and the limit laws to evaluate each limit.

$\lim_{x\to 6}\left(f(x)+\frac{1}{3}g(x)\right)$

In the following exercises, assume that $\lim_{x\to 6}f(x)=4,\lim_{x\to 6}g(x)=9,$ and $\lim_{x\to 6}h(x)=6.$ Use these three facts and the limit laws to evaluate each limit.

$\lim_{x\to 6}\frac{{(h(x))}^{3}}{2}$

In the following exercises, assume that $\lim_{x\to 6}f(x)=4,\lim_{x\to 6}g(x)=9,$ and $\lim_{x\to 6}h(x)=6.$ Use these three facts and the limit laws to evaluate each limit.

$\lim_{x\to 6}\sqrt{g(x)-f(x)}$

In the following exercises, assume that $\lim_{x\to 6}f(x)=4,\lim_{x\to 6}g(x)=9,$ and $\lim_{x\to 6}h(x)=6.$ Use these three facts and the limit laws to evaluate each limit.

$\lim_{x\to 6}x\cdot h(x)$

In the following exercises, assume that $\lim_{x\to 6}f(x)=4,\lim_{x\to 6}g(x)=9,$ and $\lim_{x\to 6}h(x)=6.$ Use these three facts and the limit laws to evaluate each limit.

$\lim_{x\to 6}[(x+1)\cdot f(x)]$

In the following exercises, assume that $\lim_{x\to 6}f(x)=4,\lim_{x\to 6}g(x)=9,$ and $\lim_{x\to 6}h(x)=6.$ Use these three facts and the limit laws to evaluate each limit.

$\lim_{x\to 6}(f(x)\cdot g(x)-h(x))$

[T] The density of an object is given by its mass divided by its volume: $\rho =m/V.$

a. Use a calculator to plot the volume as a function of density $(V=m/\rho ),$ assuming you are examining something of mass 8 kg ($m=8).$ b. Evaluate $\lim_{\rho \to {0}^{+}}V(\rho )$ and explain the physical meaning.