12.2 Finding limits: properties of limits  (Page 4/5)

Give an example of a type of function $f$ whose limit, as $x$ approaches $a,$ is $f\left(a\right).$

When direct substitution is used to evaluate the limit of a rational function as $x$ approaches $a$ and the result is $f\left(a\right)=\frac{0}{0},$ does this mean that the limit of $f$ does not exist?

What does it mean to say the limit of $f\left(x\right),$ as $x$ approaches $c,$ is undefined?

$\underset{x\to 0}{\lim }\left(3\right)$

$\underset{x\to 2}{\lim }\left(\frac{-5x}{x^2-1}\right)$

$\underset{x\to 2}{\lim }\left(\frac{x^2-5x+6}{x+2}\right)$

$\underset{x\to 3}{\lim }\left(\frac{x^2-9}{x-3}\right)$

$\underset{x\to -1}{\lim }\left(\frac{x^2-2x-3}{x+1}\right)$

$\underset{x\to \frac{3}{2}}{\lim }\left(\frac{6x^2-17x+12}{2x-3}\right)$

$\underset{x\to -\frac{7}{2}}{\lim }\left(\frac{8x^2+18x-35}{2x+7}\right)$

$\underset{x\to 3}{\lim }\left(\frac{x^2-9}{x-5x+6}\right)$

$\underset{x\to -3}{\lim }\left(\frac{-7x^4-21x^3}{-12x^4+108x^2}\right)$

$\underset{x\to 3}{\lim }\left(\frac{x^2+2x-3}{x-3}\right)$

$\underset{h\to 0}{\lim }\left(\frac{{\left(3+h\right)}^3-27}{h}\right)$

$\underset{h\to 0}{\lim }\left(\frac{{\left(2-h\right)}^3-8}{h}\right)$

$\underset{h\to 0}{\lim }\left(\frac{{\left(h+3\right)}^2-9}{h}\right)$

$\underset{h\to 0}{\lim }\left(\frac{\sqrt{5-h}-\sqrt{5}}{h}\right)$

$\underset{x\to 0}{\lim }\left(\frac{\sqrt{3-x}-\sqrt{3}}{x}\right)$

$\underset{x\to 9}{\lim }\left(\frac{x^2-81}{3-\sqrt{x}}\right)$

$\underset{x\to 1}{\lim }\left(\frac{\sqrt{x}-x^2}{1-\sqrt{x}}\right)$

$\underset{x\to 0}{\lim }\left(\frac{x}{\sqrt{1+2x}-1}\right)$

$\underset{x\to \frac{1}{2}}{\lim }\left(\frac{x^2-\frac{1}{4}}{2x-1}\right)$

$\underset{x\to 4}{\lim }\left(\frac{x^3-64}{x^2-16}\right)$

$\underset{x\to 2^{-}}{\lim }\left(\frac{|x-2|}{x-2}\right)$

$\underset{x\to 2^{+}}{\lim }\left(\frac{\left|x-2\right|}{x-2}\right)$

$\underset{x\to 2}{\lim }\left(\frac{\left|x-2\right|}{x-2}\right)$

$\underset{x\to 4^{-}}{\lim }\left(\frac{\left|x-4\right|}{4-x}\right)$

$\underset{x\to 4^{+}}{\lim }\left(\frac{\left|x-4\right|}{4-x}\right)$

$\underset{x\to 4}{\lim }\left(\frac{\left|x-4\right|}{4-x}\right)$

$\underset{x\to 2}{\lim }\left(\frac{-8+6x-x^2}{x-2}\right)$

$\underset{x\to c}{\lim }\left[2f(x)+\sqrt{g(x)}\right]$

$\underset{x\to c}{\lim }\left[3f(x)+\sqrt{g(x)}\right]$

$\underset{x\to c}{\lim }\frac{f(x)}{g(x)}$

$\underset{x\to 2}{\lim }\cos \left(\pi x\right)$

$\underset{x\to 2}{\lim }\sin \left(\pi x\right)$

$\underset{x\to 2}{\lim }\sin \left(\frac{\pi }{x}\right)$

$f(x)=\{\begin{array}{ll}2x^2+2x+1,\hfill & x\le 0\hfill \\ x-3, \hfill & x>0\hfill \end{array}\text{; }\underset{x\to 0^{+}}{\lim }f(x)$

$f(x)=\{\begin{array}{ll}2x^2+2x+1,\hfill & x\le 0\hfill \\ x-3, \hfill & x>0\hfill \end{array}\text{; }\underset{x\to 0^{-}}{\lim }f(x)$

$f(x)=\{\begin{array}{ll}2x^2+2x+1,\hfill & x\le 0\hfill \\ x-3, \hfill & x>0\hfill \end{array}\text{; }\underset{x\to 0}{\lim }f(x)$

$\underset{x\to 4}{\lim }\frac{\sqrt{x+5}-3}{x-4}$

$\underset{x\to 2^{+}}{\lim }(2x-\mathrm{〚x〛})$

$\underset{x\to 2}{\lim }\frac{\sqrt{x+7}-3}{x^2-x-2}$

$\underset{x\to 3^{+}}{\lim }\frac{x^2}{x^2-9}$

$f(x)=x+1$

$f(x)=2x^2-1$

$f(x)=x^2+3x+4$

$f(x)=x^2+4x-100$

$f(x)=3x^2+1$

$f(x)=\cos (x)$

$f(x)=2x^3-4x$

$f(x)=\frac{1}{x}$

$f(x)=\frac{1}{x^2}$

$f(x)=\sqrt{x}$

Find an equation that could be represented by [link] .

Find an equation that could be represented by [link] .

What is the right-hand limit of the function as $x$ approaches 0?

What is the left-hand limit of the function as $x$ approaches 0?

The position function $s(t)=-16t^2+144t$ gives the position of a projectile as a function of time. Find the average velocity (average rate of change) on the interval $\left[1,2\right]$ .

The height of a projectile is given by $s(t)=-64t^2+192t$ Find the average rate of change of the height from $t=1$ second to $t=1.5$ seconds.

The amount of money in an account after $t$ years compounded continuously at 4.25% interest is given by the formula $A=A_0{e}^{0.0425t},$ where $A_0$ is the initial amount invested. Find the average rate of change of the balance of the account from $t=1$ year to $t=2$ years if the initial amount invested is $1,000.00.