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Section 5.3 Limits Quiz
Subsection 5.3.1 Evaluating Limits
Checkpoint 5.3.1 .
Evaluate:
\(\lim_{x \to 2} (3x^2 - 5x + 2)\)
Checkpoint 5.3.2 .
Evaluate:
\(\lim_{x \to 3} \frac{x^2 - 9}{x - 3}\)
Checkpoint 5.3.3 .
Evaluate:
\(\lim_{x \to \infty} \frac{5x^2 + 1}{2x^2 - x}\)
Checkpoint 5.3.4 .
Evaluate:
\(\lim_{x \to \frac{\pi}{2}} \frac{\tan 2x}{x - \frac{\pi}{2}}\)
Checkpoint 5.3.5 .
Evaluate:
\(\lim_{x \to 0} (\csc x - \cot x)\)
Checkpoint 5.3.6 .
Evaluate:
\(\lim_{x \to 0} \frac{\cos 2x - 1}{\cos x - 1}\)
Checkpoint 5.3.7 .
Evaluate:
\(\lim_{x \to \pi} \frac{\sin(\pi - x)}{\pi (\pi - x)}\)
Checkpoint 5.3.8 .
Evaluate:
\(\lim_{x \to 0} \frac{\sin ax}{bx}\)
Checkpoint 5.3.9 .
Evaluate:
\(\lim_{x \to 0} \frac{(x+1)^5 - 1}{x}\)
Checkpoint 5.3.10 .
Evaluate:
\(\lim_{z \to 1} \frac{z^{1/3} - 1}{z^{1/6} - 1}\)
Subsection 5.3.2 Pieceweise Functions in Limits
Checkpoint 5.3.11 .
Evaluate the limit of the function \(f(x)\) as \(x \to 1^-\text{:}\)
\begin{equation*}
f(x) = \begin{cases}
2x + 3, & x \le 1 \\
5x - 1, & x \gt 1
\end{cases}
\end{equation*}
Checkpoint 5.3.12 .
Find \(\lim_{x \to 2^+} f(x)\) for the function:
\begin{equation*}
f(x) = \begin{cases}
x^2, & x \lt 2 \\
3x - 2, & x \ge 2
\end{cases}
\end{equation*}
Checkpoint 5.3.13 .
Determine if \(\lim_{x \to 0} f(x)\) exists for:
\begin{equation*}
f(x) = \begin{cases}
e^x, & x \lt 0 \\
x + 1, & x \ge 0
\end{cases}
\end{equation*}
No, because there is a jump
No, because e^x is undefined
Checkpoint 5.3.14 .
Find the right-hand limit as \(x \to -3^+\text{:}\)
\begin{equation*}
f(x) = \begin{cases}
|x|, & x \le -3 \\
\sqrt{x+7}, & x \gt -3
\end{cases}
\end{equation*}
Checkpoint 5.3.15 .
What value of \(k\) would make the limit exist at \(x = 4\text{?}\)
\begin{equation*}
f(x) = \begin{cases}
2x + k, & x \lt 4 \\
x^2 - 10, & x \ge 4
\end{cases}
\end{equation*}
Checkpoint 5.3.16 .
Evaluate \(\lim_{x \to 1} f(x)\) if:
\begin{equation*}
f(x) = \begin{cases}
x^2 - 4x, & x \lt 1 \\
-3, & x = 1 \\
2x - 5, & x \gt 1
\end{cases}
\end{equation*}
Checkpoint 5.3.17 .
Identify the type of discontinuity at \(x = 5\text{:}\)
\begin{equation*}
f(x) = \begin{cases}
\frac{1}{x-5}, & x \lt 5 \\
x^2, & x \ge 5
\end{cases}
\end{equation*}
Checkpoint 5.3.18 .
Find \(\lim_{x \to -1^-} f(x)\) for:
\begin{equation*}
f(x) = \begin{cases}
5, & x \le -1 \\
x + 6, & x \gt -1
\end{cases}
\end{equation*}
Checkpoint 5.3.19 .
Given
\(f(x) = \begin{cases} x^2 - 1, & x \le 1 \\ -x^2 - 1, & x \gt 1 \end{cases}\text{,}\) what is the exact value of
\(f(1)\text{?}\)
Checkpoint 5.3.20 .
If
\(\lim_{x \to a^-} f(x) = L\) and
\(\lim_{x \to a^+} f(x) = M\text{,}\) the two-sided limit exists if and only if:
Subsection 5.3.3 Points of Discontinuity in Limits
Checkpoint 5.3.21 .
Identify the point of discontinuity for the function \(f(x) = \frac{x^2 - 9}{x - 3}\).
Checkpoint 5.3.22 .
Where is the function \(f(x) = \frac{1}{x^2 - 4}\) discontinuous?
Checkpoint 5.3.23 .
What type of discontinuity exists at x = 0 for \(f(x) = \frac{\sin(x)}{x}\)?
The function is continuous
Checkpoint 5.3.24 .
For
\(f(x) = \frac{x+2}{x^2+5x+6}\text{,}\) identify the vertical asymptote(s).
Checkpoint 5.3.25 .
A jump discontinuity occurs when:
The left and right hand limits are equal but do not equal the function value.
The limit goes to infinity.
The left and right hand limits exist but are not equal.
The function is undefined at that point only.
Checkpoint 5.3.26 .
Identify the point of discontinuity for
\(f(x) = \ln(x)\text{.}\)
There are no points of discontinuity in its domain.
Checkpoint 5.3.27 .
Find the value of \(k\) that makes \(f(x)\) continuous at x = 2:
\(f(x) = \begin{cases} x^2 & \text{if } x \le 2 \\ kx & \text{if } x & \gt; 2 \end{cases} \text{.}\)
Checkpoint 5.3.28 .
Which function has an infinite discontinuity at x = 0?
Checkpoint 5.3.29 .
The function
\(f(x) = \frac{x-1}{x^2-1}\) has:
A hole at x = 1 and asymptote at x = -1
A hole at x = -1 and asymptote at x = 1
Checkpoint 5.3.30 .
If
\(\lim_{x \to c} f(x)\) does not exist, the function is definitely discontinuous at c.
