Determine whether the following statements are true and give an explanation or counterexample.


a. The value of $\underset{x\to 3}{\lim }\frac{x^2-9}{x-3}$ does not exist.

Let $g\(\left\)(x\(\right\))=\(\frac{x^3-4x}{8\left|x-2\right|}\)$. <IMAGE>

Calculate $g\(\left\)(x\(\right\))$ for each value of $x$ in the following table.

Let $g\(\left\)(x\(\right\))=\(\frac{x^3-4x}{8\left|x-2\right|}\)$. <IMAGE>

Make a conjecture about the values of ${\(\displaystyle\]\lim\)_{x\(\to\)2^{-}}g\(\left\)(x\(\right\))}$, ${\(\displaystyle\]\lim\)_{x\(\to\)2^{+}}g\(\left\)(x\(\right\))}$, and ${\(\displaystyle\]\lim\)_{x\(\to\)2}g\(\left\)(x\(\right\))}$ or state that they do not exist.

Use a graph of f to estimate $\underset{x\to a}{\lim }f\left(x\right)$ or to show that the limit does not exist. Evaluate f(x) near $x=a$ to support your conjecture.

$f\left(x\right)=\frac{x-2}{\ln \mid x-2\mid }$; $a=2$

Use a graph of f to estimate $\underset{x\to a}{\lim }f\left(x\right)$ or to show that the limit does not exist. Evaluate f(x) near $x=a$ to support your conjecture.

$f\left(x\right)=\frac{1-\cos (2x-2)}{{(x-1)}^2};a=1$

Determine whether the following statements are true and give an explanation or counterexample.

d. $\underset{x\to 0}{\lim }\sqrt{x}$ . (Hint: Graph y=√x)

Determine whether the following statements are true and give an explanation or counterexample.

e. $\underset{x\to \frac{\pi }{2}}{\lim }\cot x=0$ . (Hint: Graph y=cot x)

Sketch the graph of a function with the given properties. You do not need to find a formula for the function. 

f(2) = 1,lim x→2 f(x) = 3

Sketch the graph of a function with the given properties. You do not need to find a formula for the function. 

p(0) = 2,lim x→0 p(x) = 0,lim x→2 p(x) does not exist, p(2)=lim x→2^+ p(x)=1