Question

Find the limit. $$\lim _{x \rightarrow 0} \frac{\sin 4 x}{\sin 6 x}$$

   Find the limit.
$$\lim _{x \rightarrow 0} \frac{\sin 4 x}{\sin 6 x}$$

Essential Calculus Early Transcendentals

Essential Calculus Early Transcendentals

James Stewart 2nd Edition

Chapter 1, Problem 50

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Solved by Expert Yaw Asomani

05/27/2020

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This gives us: $$\lim _{x \rightarrow 0} \frac{\sin 4 x \cdot 4x}{\sin 6 x \cdot 6x}$$ Show more…

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Find the limit. $$\lim _{x \rightarrow 0} \frac{\sin 4 x}{\sin 6 x}$$

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Key Concepts

Trigonometric Limits

This concept involves evaluating limits that include trigonometric functions, particularly using standard limits such as lim (x -> 0) [sin(x)/x] = 1. Understanding these limits is essential for handling ratios of sine functions as the variable approaches 0, a common scenario in calculus.

Small-Angle Approximation

This principle states that for small angles measured in radians, the sine of the angle is approximately equal to the angle itself. This approximation simplifies the analysis of trigonometric limits by reducing complex expressions to more manageable linear forms.

Indeterminate Forms

Indeterminate forms occur when the direct substitution in a limit problem yields an undefined expression like 0/0. Recognizing an indeterminate form is crucial as it signals the need for additional techniques such as algebraic manipulation or L’Hôpital’s rule to accurately evaluate the limit.

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