1. Limits and Continuity

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.

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

For any real number x, the floor function (or greatest integer function) ⌊x⌋ is the greatest integer less than or equal to x (see figure).

a. Compute lim x→−1^− ⌊x⌋, lim x→−1^+ ⌊x⌋,lim x→2^− ⌊x⌋, and lim x→2^+ ⌊x⌋.

A function f is even if f(−x)=f(x), for all x in the domain of f. Suppose f is even, with lim x→2^+ f(x)=5 and lim x→2^− f(x)=8. Evaluate the following limits.

a. lim x→−2^+ f(x)

Estimate the following limits using graphs or tables.

$\underset{h\to 0}{\lim }\frac{\ln (1+h)}{h}$

Estimate the following limits using graphs or tables.

lim x→1 9(√2x − x^4 −3√x) / 1 − x^3/4

Determine ${\lim }_{x\to \infty }f\left(x\right)$ and ${\lim }_{x\to -\infty }f\left(x\right)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).

$f\left(x\right)=\frac{4x}{20x+1}$

Determine $\lim_{x\rightarrow\infty}f\left(x\right)$ and $\lim_{x\rightarrow-\infty}f\left(x\right)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).

$f\left(x\right)=\frac{6x^2-9x+8}{3x^2+2}$

Determine $\lim_{x\rightarrow\infty}f\left(x\right)$ and $\lim_{x\rightarrow-\infty}f\left(x\right)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).

$f\left(x\right)=\frac{3x^3-7}{x^4+5x^2}$

Determine $\lim_{x\rightarrow\infty}f\left(x\right)$ and $\lim_{x\rightarrow-\infty}f\left(x\right)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).

$f\left(x\right)=\frac{40x^5+x^2}{16x^4-2x}$

Determine $\lim_{x\rightarrow\infty}f\left(x\right)$ and $\lim_{x\rightarrow-\infty}f\left(x\right)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).

$f\left(x\right)=\frac{1}{2x^4-\sqrt{4x^8-9x^4}}$

Determine $\lim_{x\rightarrow\infty}f\left(x\right)$ and $\lim_{x\rightarrow-\infty}f\left(x\right)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).

$f\left(x\right)=\frac{4x^3+1}{2x^3+\sqrt{16x^6+1}}$