Q: Explain why the following limit does not exist: $$\lim\limits_{x \to 0} \frac{1}{x}.$$

Because the left and right limits as $$x \to 0$$ are not equal; one approaches $$-\infty$$ and the other $$+\infty.$$

Question 1

Use the precise definition of limit to find a value of $$\delta$$ such that $$|\sqrt{x^2$+7} - 3| \u003c 0.5$$ whenever $$|x - 2| \u003c \delta.$$

Accepted Answer

$$\delta = 1.4$$

Question 2

Evaluate the following limit: $$\lim\limits_{x 	o 0} \frac{\sin(4x)}{x}.$$

Accepted Answer

$$4$$

Question 3

Explain why the following limit does not exist: $$\lim\limits_{x 	o 0} \frac{1}{x}.$$

Accepted Answer

Because the left and right limits as $$x 	o 0$$ are not equal; one approaches $$-\infty$$ and the other $$+\infty.$$

Question 4

Find $$\lim\limits_{x 	o 2} \frac{x^2$ - 4}{x - 2}.$$

Accepted Answer

$$4$$

Question 5

Evaluate $$\lim\limits_{x 	o 0} \frac{\sin(2x)}{x}.$$

Accepted Answer

$$2$$

Question 6

Find $$\lim\limits_{x 	o 0} \frac{\cos(x) - 1}{x}.$$

Accepted Answer

$$0$$

Question 7

Let $$f(x) = \begin{cases} \sqrt{x}, & x \geq 0 \ 1 - \cos x, & x \u003c 0 \end{cases}. $$Which of the following is true?

Accepted Answer

$$f(x) is $$continuous at $$x = 0.$$

Question 8

Given $$y(x) = \frac{2x^2$}{x^2$ + 1}$$, find the slope of the tangent line to the curve at the point $$(3, 3).$$

Accepted Answer

$$\frac{12}{5}$$

Question 9

Use the definition of derivative to find $$f'(x)$$ for $$f(x) = \frac{1}{x}.$$

Accepted Answer

$$-\frac{1}{x^2$}$$

Question 10

A ball is thrown vertically upward with a velocity of $$20 m/s $$from a height of $$10 m. $$Its height after $$t$$ seconds is $$h(t) = -4.9t^2$ + 20t + 10. $$What is the instantaneous velocity after $$2$$ seconds?

Accepted Answer

Question 11

Evaluate $$\lim\limits_{x 	o 0} \frac{\sin(3x)}{x}.$$

Accepted Answer

$$3$$

Question 12

Find the derivative of $$f(x) = x^2$ \sin x.$$

Accepted Answer

$$2x \sin x + x^2$ \cos x$$

Question 13

If $$f(x) = \ln(x^2$ + 1)$$, what is $$f'(x)$$?

Accepted Answer

$$\frac{2x}{x^2$ + 1}$$

Question 14

A medication is administered and its concentration in the bloodstream is modeled by $$C(t) = 5e$$^{-0.2t}$$, where t$ is in hours. What is the rate of change of concentration after 3$ hours?

Accepted Answer

$$-1e^{-0.6}$$

Question 15

Evaluate $$\lim\limits_{x 	o 0} \frac{1 - \cos(5x)}{x^2$}.$$

Accepted Answer

$$\frac{25}{2}$$

Question 16

Find the derivative of $$f(x) = 	an$$^{-1}$$(x).$$

Accepted Answer

$$\frac{1}{1 + x^2$}$$

Question 17

Suppose $$f(x) is $$continuous everywhere and $$f(2) = 5$$, $$f'(2) = -3. $$Find the equation of the tangent line to $$f(x) at$$ $$x = 2.$$

Accepted Answer

$$y = -3(x - 2) + 5$$

Question 18

If $$f(x) = x^3$ - 2x + 1$$, find $$f'(1).$$

Accepted Answer

$$3$$

Question 19

A bacteria population grows according to $$P(t) = 100e$$^{0.05t}$$. What is the instantaneous rate of growth at t = 10$ hours?

Accepted Answer

$$5e^{0.5}$$

Question 20

Evaluate $$\lim\limits_{x 	o 0} \frac{e$$^{2x}$$ - 1}{x}.$$

Accepted Answer

$$2$$

Calculus Study Notes: Limits and Derivatives (Exam Practice)

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