Gilbert Strang; Edwin “Jed” Herman

Review Exercises

True or False. In the following exercises, justify your answer with a proof or a counterexample.

208.

A function has to be continuous at $x = a$ if the $lim_{x \to a} f (x)$ exists.

209.

You can use the quotient rule to evaluate $lim_{x \to 0} \frac{sin x}{x} .$

210.

If there is a vertical asymptote at $x = a$ for the function $f (x),$ then f is undefined at the point $x = a .$

211.

If $lim_{x \to a} f (x)$ does not exist, then f is undefined at the point $x = a .$

212.

Using the graph, find each limit or explain why the limit does not exist.

$lim_{x \to −1} f (x)$
$lim_{x \to 1} f (x)$
$lim_{x \to 0^{+}} f (x)$
$lim_{x \to 2} f (x)$

A graph of a piecewise function with several segments. The first is a decreasing concave up curve existing for x < -1. It ends at an open circle at (-1, 1). The second is an increasing linear function starting at (-1, -2) and ending at (0,-1). The third is an increasing concave down curve existing from an open circle at (0,0) to an open circle at (1,1). The fourth is a closed circle at (1,-1). The fifth is a line with no slope existing for x > 1, starting at the open circle at (1,1).

In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.

213.

$lim_{x \to 2} \frac{2 x^{2} - 3 x - 2}{x - 2}$

214.

$lim_{x \to 0} 3 x^{2} - 2 x + 4$

215.

$lim_{x \to 3} \frac{x^{3} - 2 x^{2} - 1}{3 x - 2}$

216.

$lim_{x \to π / 2} \frac{cot x}{cos x}$

217.

$lim_{x \to −5} \frac{x^{2} + 25}{x + 5}$

218.

$lim_{x \to 2} \frac{3 x^{2} - 2 x - 8}{x^{2} - 4}$

219.

$lim_{x \to 1} \frac{x^{2} - 1}{x^{3} - 1}$

220.

$lim_{x \to 1} \frac{x^{2} - 1}{\sqrt{x} - 1}$

221.

$lim_{x \to 4} \frac{4 - x}{\sqrt{x} - 2}$

222.

$lim_{x \to 4} \frac{1}{\sqrt{x} - 2}$

In the following exercises, use the squeeze theorem to prove the limit.

223.

$lim_{x \to 0} x^{2} cos (2 π x) = 0$

224.

$lim_{x \to 0} x^{3} sin (\frac{π}{x}) = 0$

225.

Determine the domain such that the function $f (x) = \sqrt{x - 2} + x e^{x}$ is continuous over its domain.

In the following exercises, determine the value of c such that the function remains continuous. Draw your resulting function to ensure it is continuous.

226.

$f (x) = {\begin{cases} x^{2} + 1, x > c \\ 2 x, x \leq c \end{cases}$

227.

$f (x) = {\begin{cases} \sqrt{x + 1}, x > − 1 \\ x^{2} + c, x \leq - 1 \end{cases}$

In the following exercises, use the precise definition of limit to prove the limit.

228.

$lim_{x \to 1} (8 x + 16) = 24$

229.

$lim_{x \to 0} x^{3} = 0$

230.

A ball is thrown into the air and the vertical position is given by $x (t) = −4.9 t^{2} + 25 t + 5 .$ Use the Intermediate Value Theorem to show that the ball must land on the ground sometime between 5 sec and 6 sec after the throw.

231.

A particle moving along a line has a displacement according to the function $x (t) = t^{2} - 2 t + 4,$ where x is measured in meters and t is measured in seconds. Find the average velocity over the time period $t = [0, 2] .$

232.

From the previous exercises, estimate the instantaneous velocity at $t = 2$ by checking the average velocity within $t = 0.01 sec .$