Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
88. lim(x→0) (tan x)/(x + sin(x))

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First, identify the form of the limit as \( x \to 0 \). Substitute \( x = 0 \) into the expression \( \frac{\tan x}{x + \sin x} \) to check if it results in an indeterminate form.

Since \( \tan 0 = 0 \), \( 0 + \sin 0 = 0 \), the limit is of the form \( \frac{0}{0} \), which is indeterminate and allows the use of l’Hôpital’s Rule.

Apply l’Hôpital’s Rule by differentiating the numerator and denominator separately with respect to \( x \). The derivative of the numerator \( \tan x \) is \( \sec^2 x \).

The derivative of the denominator \( x + \sin x \) is \( 1 + \cos x \).

Rewrite the limit as \( \lim_{x \to 0} \frac{\sec^2 x}{1 + \cos x} \) and then evaluate this new limit by substituting \( x = 0 \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

l’Hôpital’s Rule

l’Hôpital’s Rule is a method for evaluating limits that result in indeterminate forms like 0/0 or ∞/∞. It states that the limit of a ratio of functions can be found by taking the limit of the ratio of their derivatives, provided certain conditions are met.

Limit of a Function as x Approaches a Point

The limit describes the value that a function approaches as the input approaches a specific point. Understanding how to evaluate limits, especially when direct substitution leads to indeterminate forms, is essential for applying techniques like l’Hôpital’s Rule.

Derivatives of Trigonometric Functions

Knowing the derivatives of trigonometric functions such as tan(x) and sin(x) is crucial when applying l’Hôpital’s Rule. For example, the derivative of tan(x) is sec²(x), and the derivative of sin(x) is cos(x), which are used to find the new limit after differentiation.

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