Using the standard limit $$\lim\limits_{x \to 0} \frac{\sin x}{x} = 1$$

Question 1

Which trigonometric identity is used in the solution to express $$\cos x in $$terms of $$\sin^2$ \frac{x}{2}$$?

Accepted Answer

$$\cos x = 1 - 2 \sin^2$ \frac{x}{2}$$

Question 2

Evaluate $$\lim\limits_{x 	o 0} \frac{-2 \sin^2$ \frac{x}{2}}{x}$$ using the double angle identity.

Accepted Answer

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

Question 3

Which limit property is used to separate $$\lim\limits_{x 	o 0} 2 \cdot \frac{\sin \frac{x}{2}}{x} \cdot \sin \frac{x}{2}$$ into two limits?

Accepted Answer

Product of limits

Question 4

What is the value of $$\lim\limits_{x 	o 0} \frac{\sin \frac{x}{2}}{\frac{x}{2}}$$?

Accepted Answer

$$1$$

Question 5

In the context of the limit $$\lim\limits_{x 	o 0} \frac{-2 \sin^2$ \frac{x}{2}}{x}$$, what common calculus technique is used to evaluate the limit?

Accepted Answer

Using the standard limit $$\lim\limits_{x 	o 0} \frac{\sin x}{x} = 1$$

Question 6

Suppose you are asked to evaluate $$\lim\limits_{x 	o 0} \frac{\sin^2$ x}{x}. $$Which substitution would simplify the limit?

Accepted Answer

Let $$u = x$$

Question 7

If $$\lim\limits_{x 	o 0} \frac{\sin x}{x} = 1$$, what is $$\lim\limits_{x 	o 0} \frac{\sin^2$ x}{x^2$}$$?

Accepted Answer

$$1$$

Question 8

Which of the following best describes the real-world application of evaluating trigonometric limits as $$x 	o 0$$?

Accepted Answer

Calculating instantaneous velocity in physics

Question 9

Given $$\lim\limits_{x 	o 0} \frac{\sin ax}{x} = a$$, what is $$\lim\limits_{x 	o 0} \frac{\sin 3x}{x}$$?

Accepted Answer

$$3$$

Question 10

Which step is necessary to evaluate $$\lim\limits_{x 	o 0} \frac{\sin^2$ \frac{x}{2}}{x}$$?

Accepted Answer

Rewrite $$\sin^2$ \frac{x}{2} as$$ $$\left(\sin \frac{x}{2}\$$right)^2$$ and use the limit property

Question 11

If $$\lim\limits_{x 	o 0} \frac{\sin x}{x} = 1$$, what is $$\lim\limits_{x 	o 0} \frac{\sin \frac{x}{2}}{x}$$?

Accepted Answer

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

Question 12

Which of the following is a correct manipulation to evaluate $$\lim\limits_{x 	o 0} \frac{\sin^2$ \frac{x}{2}}{x}$$?

Accepted Answer

Write $$\frac{\sin^2$ \frac{x}{2}}{x} = \frac{\sin \frac{x}{2}}{\frac{x}{2}} \cdot \frac{\sin \frac{x}{2}}{2}$$

Limits Involving Trigonometric Functions and Their Applications

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