AP Calculus AB: 1.10 Exploring Types of 3 Discontinuities - Exam Style questions with Answer- MCQ

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Question

Let f be the function given by $f(x)=\frac{x-2}{2\left | x-2 \right |}$. Which of the following is true?

A $\lim_{x\rightarrow 2}f(x)=\frac{1}{2}$

B f has a removable discontinuity at x = 2.

C f has a jump discontinuity at x = 2

D f has a discontinuity due to a vertical asymptote at x = 2.

▶️Answer/Explanation

Ans:C

Question

Let f be the function given by $f(x)=\frac{x-2}{2\left | x-2 \right |}$.Which of the following is true?

A $\lim_{x\rightarrow 2}f(x)=\frac{1}{2}$

B f has a removable discontinuity at x = 2.

C f has a jump discontinuity at x = 2 .

D f has a discontinuity due to a vertical asymptote at x = 2.

▶️Answer/Explanation

Ans:C

Question

Let $f$

$f$

be the function defined by $f(x)=\frac{3x^{3}+x^{2}}{x^{2}-x}$.Which of the following statements is true?

$f$ has a discontinuity due to a vertical asymptote at $x equals 0$ and at $x equals 1$ .

B $f$ has a removable discontinuity at $x equals 0$ and a jump discontinuity at $x equals 1$ .

C $f$ has a removable discontinuity at $x equals 0$ and a discontinuity due to a vertical asymptote at $x equals 1$ .

D $f$ is continuous at $x equals 0$ , and $f$ has a discontinuity due to a vertical asymptote at $x equals 1$

▶️Answer/Explanation

.Ans:C

The function $f$ is not defined at $x equals 0$ because the denominator equals 0 when $x equals 0$ . However, $\lim_{x\rightarrow 0}f(x)$ exists, as shown below. Therefore, $f$ has a removable discontinuity at $x equals 0$

$\lim_{x\rightarrow 0}f(x)=\lim_{x\rightarrow 0}\frac{3x^{3}+2x^{2}}{x^{2}-x}=\lim_{x\rightarrow 0}\frac{x(3x^{2}+2x)}{x(x-1)}$ = $\lim_{x\rightarrow 0}\frac{3x^{2}+2x}{x-1}=\frac{3.0^{2}+2.0}{0-1}=0$

The graph of the rational function $f$

f

has a vertical asymptote at $x equals 1$ because the numerator is nonzero and the denominator equals 0 when $x equals 1$

Question

The graph of the function $f$

$f$

is shown above. What are all values of $x$ for which $f$ has a removable discontinuity?

A 0 only

B 1 only

C 0 and 2 only

D 0,1 and 2

▶️Answer/Explanation

Ans:C

A removable discontinuity occurs at x=c, if $\lim_{x\rightarrow c}f(x)=L$ ,exists but $f (c)$ is not defined or $f (c) \neq L$

L

. For this function , $\lim_{x\rightarrow 0}f(x)$ exists but $f of 0$ is not defined, and $\lim_{x\rightarrow 2}f(x)$ exists but $f of 2$ is not defined. Therefore, there are removable discontinuities at $x equals 0$ and at $x equals 2$ .

AP Calculus AB: 1.10 Exploring Types of 3 Discontinuities – Exam Style questions with Answer- MCQ

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