Question 1

Given the function $$f(x) = \frac{x^2$ - 1}{x - 1}$$, what is $$\lim_{x 	o 1} f(x)$$?

Accepted Answer

$$2$$

Question 2

Which of the following is the correct application of the Squeeze Theorem to evaluate $$\lim_{x 	o 0} x^2$ \sin\left(\frac{1}{x}ight)$$?

Accepted Answer

Since $$-x^2$$ \leq $$x^2$$ \sin\left(\frac{1}{x}\right) \leq $$x^2$$, and both bounds go to $$0$$, the limit is $$0$$

Question 3

Evaluate $$\lim_{x 	o \infty} \frac{3x^2$ + 2x}{x^2$ - x}.$$

Accepted Answer

$$3$$

Question 4

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

Accepted Answer

$$4$$

Question 5

Using the identity $$\sin^2$(x) + \cos^2$(x) = 1$$, what is $$\lim_{x 	o 0} \frac{1 - \cos^2$(x)}{x^2$}$$?

Accepted Answer

$$1$$

Question 6

Suppose the graph of $$f(x)$$ approaches $$3 as$$ $$x$$ approaches $$-2$$ from both sides. What is $$\lim_{x 	o -2} f(x)$$?

Accepted Answer

$$3$$

Question 7

Evaluate $$\lim_{x 	o 0} \frac{\sin(5x)}{x}.$$

Accepted Answer

$$5$$

Question 8

Which of the following statements is true about $$\lim_{x 	o \infty} \frac{1}{x}$$?

Accepted Answer

The limit is $$0$$

Question 9

A chemical reaction rate is modeled by $$R(t) = \frac{10t}{t^2$ + 1}. $$What is $$\lim_{t 	o \infty} R(t)$$?

Accepted Answer

$$0$$

Question 10

Evaluate $$\lim_{x 	o 0} \frac{1 - \cos(x)}{x^2$}$$ using trigonometric identities.

Accepted Answer

$$\frac{1}{2}$$

Question 11

Given $$f(x) = \begin{cases} 2x + 1 & x \u003c 1 \ x^2$ & x \geq 1 \end{cases}$$, what is $$\lim_{x 	o 1^+} f(x)$$?

Accepted Answer

$$1$$

Question 12

Which technique is most appropriate to evaluate $$\lim_{x 	o 2} \frac{x^2$ - 4}{x - 2}$$?

Accepted Answer

Factoring

Question 13

A population model is given by $$P(t) = \frac{1000}{1 + e$$^{-t}$$}. $$What is $$\lim_{t 	o \infty} P(t)$$?

Accepted Answer

$$1000$$

Question 14

Evaluate $$\lim_{x 	o 0} \frac{\sin^2$(x)}{x^2$}.$$

Accepted Answer

$$1$$

Question 15

Which of the following is the correct left-hand limit for $$f(x) = |x| as$$ $$x 	o 0^-$$?

Accepted Answer

$$0$$

Question 16

A medication concentration in blood is modeled by $$C(t) = \frac{5}{t}$$, where $$t is $$time in hours. What is $$\lim_{t 	o \infty} C(t)$$?

Accepted Answer

$$0$$

Question 17

Evaluate $$\lim_{x 	o 0} \frac{	an(x)}{x}.$$

Accepted Answer

$$1$$

Question 18

Which of the following is true about $$\lim_{x 	o 0} \frac{1}{x}$$?

Accepted Answer

The limit does not exist

Question 19

Evaluate $$\lim_{x 	o \infty} \frac{2x + 1}{x - 3}.$$

Accepted Answer

$$2$$

Question 20

A temperature function is $$T(t) = 20 + 5e$$^{-t}$$, where t$ is time in hours. What is $$\lim_{t 	o \infty} T(t)$$?

Accepted Answer

$$20$$

Study Guide: Limits and Techniques for Calculus Exams

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