Question 1

Suppose $$f(x) is $$defined as follows:

$$ f(x) = \begin{cases} x^2$ & x \u003c 2 \ 3x-2 & x \geq 2 \end{cases} $$

What is $$\displaystyle \lim_{x 	o 2^-} f(x)$$?

Accepted Answer

$$4$$

Question 2

Given the graph of $$f(x)$$ has a hole at $$x=1$$ but both sides approach the same height $$y=3$$, which of the following is true?

Accepted Answer

The limit exists at $$x=1$$ but $$f(1)$$ may not be defined.

Question 3

Evaluate $$\displaystyle \lim_{x 	o 0} \begin{cases} \sin(x) & x \u003c 0 \ x^2$ & x \geq 0 \end{cases}.$$

Accepted Answer

$$0$$

Question 4

A function $$g(x) is $$defined as $$g(x) = \begin{cases} 2x+1 & x \u003c 3 \ x^2$-2 & x \geq 3 \end{cases}. $$What is $$\displaystyle \lim_{x 	o 3^+} g(x)$$?

Accepted Answer

$$9$$

Question 5

Which of the following statements is TRUE about the limit $$\displaystyle \lim_{x 	o a} f(x)$$?

Accepted Answer

The limit exists if and only if the left and right limits are equal.

Question 6

Consider the piecewise function $$h(x) = \begin{cases} x+2 & x  1 \end{cases}. $$Does $$\displaystyle \lim_{x 	o 1} h(x)$$ exist?

Accepted Answer

No, because the left and right limits are not equal.

Question 7

If a function $$f(x)$$ has a jump discontinuity at $$x=4$$, which of the following is most likely TRUE?

Accepted Answer

The left and right limits at $$x=4$$ are not equal.

Question 8

A function $$p(x) is $$defined as $$p(x) = \begin{cases} x^2$ & x  0 \end{cases}. $$What is $$\displaystyle \lim_{x 	o 0} p(x)$$?

Accepted Answer

Does not exist

Question 9

Which of the following best describes a removable discontinuity?

Accepted Answer

The function is not defined at the point, but the left and right limits are equal.

Question 10

Suppose $$f(x) is $$defined as $$f(x) = \begin{cases} 2x & x  1 \end{cases}. $$What is $$\displaystyle \lim_{x 	o 1^-} f(x)$$ and $$\displaystyle \lim_{x 	o 1^+} f(x)$$?

Accepted Answer

$$2$$ and $$2$$

Question 11

A function $$q(x) is $$defined as $$q(x) = \begin{cases} x^2$ & x \u003c 2 \ 4 & x \geq 2 \end{cases}. $$What is $$\displaystyle \lim_{x 	o 2} q(x)$$?

Accepted Answer

$$4$$

Question 12

If $$\displaystyle \lim_{x 	o 3^-} f(x) = 5$$ and $$\displaystyle \lim_{x 	o 3^+} f(x) = 7$$, what can you conclude about $$\displaystyle \lim_{x 	o 3} f(x)$$?

Accepted Answer

The limit does not exist.

Question 13

Which of the following is NOT a step in evaluating the limit of a piecewise function at a boundary point?

Accepted Answer

Always use the function's value at the point.

Question 14

A real-world scenario: The temperature $$T(t) in a $$chemical reaction is modeled as $$T(t) = \begin{cases} 20 + 2t & t \u003c 5 \ 30 & t \geq 5 \end{cases}. $$What is $$\displaystyle \lim_{t 	o 5^-} T(t)$$?

Accepted Answer

$$30$$

Question 15

If a function $$f(x) is $$not defined at $$x=a$$, but $$\displaystyle \lim_{x 	o a^-} f(x) = \lim_{x 	o a^+} f(x) = L$$, what type of discontinuity is present at $$x=a$$?

Accepted Answer

Removable discontinuity

Question 16

Given $$f(x) is $$continuous everywhere except at $$x=2$$, where $$\displaystyle \lim_{x 	o 2^-} f(x) = 1$$ and $$\displaystyle \lim_{x 	o 2^+} f(x) = 3$$, what is true about $$f(x) at$$ $$x=2$$?

Accepted Answer

There is a jump discontinuity at $$x=2.$$

Question 17

Suppose $$f(x) is $$defined as $$f(x) = \begin{cases} x^2$ & x  1 \end{cases}. $$What is $$\displaystyle \lim_{x 	o 1} f(x)$$?

Accepted Answer

$$2$$

Question 18

Which of the following best describes the difference between the function value and the limit at a point?

Accepted Answer

The function value is what the function equals at the point; the limit is what the function approaches as $$x$$ approaches the point.

Limits and Graphs: Study Guide for Calculus Students

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