Gateway 1 Practice Test

NO CALCULATORS MAY BE USED ON THIS TEST.

1. (20 pts.)

Find the limits (or write DNE for does not exist) for the function y = f(x) given by the graph:

(a)

$${\lim_{x\rightarrow -2}f(x) = }$$

(b)

$${\lim_{x\rightarrow -1}f(x) = }$$

(c)

$${\lim_{x\rightarrow 1^+}f(x) = }$$

(d)

$${\lim_{x\rightarrow 1^-}f(x) = }$$

(e)

$${\lim_{x\rightarrow 1}f(x) = }$$

2. (8 pts.)

For the function y = f(x) defined on the interval [-3,3] by the graph above, at what values of x in the interval is the function not continuous?

3. (30 pts.)

Find the limits.

(a)

$${\lim_{x\rightarrow 3}x^2-x+1 }$$

(b)

$${\lim_{x\rightarrow -2}\frac{(x+2)(x+3)}{(x+2)} }$$

(c)

$${\lim_{t\rightarrow 2^-}\frac{3}{(t-2)} }$$

(d)

$${\lim_{t\rightarrow 2^+}\frac{3}{(t-2)} }$$

(e)

$${\lim_{s\rightarrow 5}\frac{5}{(s-5)^2} }$$

(f)

$${\lim_{t\rightarrow 0}\frac{5t}{\sin(2t)} }$$

4. (12 pts.)

At which values of the independent variable are the following functions continuous?

(a)

$${f(x) = 16x^4 -3x^3+2x+1 }$$

(a)

$${g(y) = \frac{6y}{y^2-9} }$$

5. (10 pts)

Find the limit: $${\lim_{x\rightarrow -\infty}\frac{1}{(x+1)^2}-2 }$$

6. (10 pts)

Find the limit: $${\lim_{u\rightarrow \infty}\frac{3u^2+1}{u^2+3} }$$

7. (10 pts)

Give the equations of the horizontal and vertical asymptotes of the given function. You should assume that the x-axis is horizontal.

$$y = \frac{3x+1}{2-x}$$

Horizontal Asymptote:

Vertical Asymptote: