Evaluate the trigonometric limit: \lim_{x \to 0} \frac{\cos(3x) - \cos(x)}{x^2} What is th… — JAMB UTME

Evaluate the trigonometric limit:

\lim_{x \to 0} \frac{\cos(3x) - \cos(x)}{x^2}

What is the numerical value of this limit?

Answer: -4

Answer

The numerical value of the limit is -4.

Using either the sum-to-product identity

\cos(3x) - \cos(x) = -2 \sin(2x) \sin(x)

along with standard limits

\lim_{x \to 0} \frac{\sin(kx)}{x} = k

, or applying L'Hôpital's rule twice on the

\frac{0}{0}

form, yields the exact value

-4

Step-by-Step Solution

Check the form of the limit by direct substitution

Substituting

x = 0

gives

\frac{\cos(0) - \cos(0)}{0^2} = \frac{1 - 1}{0} = \frac{0}{0}

, an indeterminate form.

Determines whether algebraic transformation or L'Hôpital's rule is required.

Transform the numerator using the sum-to-product formula

\cos(3x) - \cos(x) = -2 \sin\left(\frac{3x+x}{2}\right) \sin\left(\frac{3x-x}{2}\right) = -2 \sin(2x) \sin(x)

Converts difference of cosines into product of sines to utilize standard trigonometric limits.

Rewrite the fractional expression and apply limit laws

\lim_{x \to 0} \frac{-2 \sin(2x) \sin(x)}{x^2} = -2 \cdot \left(\lim_{x \to 0} \frac{\sin(2x)}{x}\right) \cdot \left(\lim_{x \to 0} \frac{\sin(x)}{x}\right)

Splits

x^2

into

x \cdot x

under each sine function.

Evaluate the individual standard limits

\lim_{x \to 0} \frac{\sin(2x)}{x} = 2 \quad \text{and} \quad \lim_{x \to 0} \frac{\sin(x)}{x} = 1

Applies the known fundamental trigonometric limit rule

\lim_{u \to 0} \frac{\sin(au)}{u} = a

Calculate the final product

-2 \times 2 \times 1 = -4

Combines all factors to reach the evaluated value.

Key Concept

Trigonometric Limits and Indeterminate Forms

Alternative Method

Alternatively, apply L'Hôpital's rule twice. First derivative of numerator over denominator yields

\lim_{x \to 0} \frac{-3\sin(3x) + \sin(x)}{2x}

(still

\frac{0}{0}

). Differentiating a second time yields

\lim_{x \to 0} \frac{-9\cos(3x) + \cos(x)}{2} = \frac{-9(1) + 1}{2} = \frac{-8}{2} = -4

Estimated Time:1m 30s

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