Problem 62 - A Graphical Approach to Precalculus with Limits

Question

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value. \(\lim _{x \rightarrow 0} \frac{2 x}{\tan x}\)

Step-by-Step Solution

Answer

The limit is 2.

Step 1: Problem Identification

We need to determine if the limit \( \lim_{x \rightarrow 0} \frac{2x}{\tan x} \) exists. If it does exist, we will find the value of this limit.

Step 2: Choosing the Right Strategy

Since direct substitution in the expression \( \frac{2x}{\tan x} \) leads to an indeterminate form \( \frac{0}{0} \), we will use alternative methods such as L'Hôpital's Rule or series expansion to evaluate the limit.

Step 3: Applying L'Hôpital's Rule

L'Hôpital's Rule is applicable for indeterminate forms like \( \frac{0}{0} \). According to the rule, if the limit \( \lim_{x \to c} \frac{f(x)}{g(x)} = \frac{0}{0} \), then it can be computed as \( \lim_{x \to c} \frac{f'(x)}{g'(x)} \), provided the new limit exists. With \( f(x) = 2x \) and \( g(x) = \tan x \), we find the derivatives: \( f'(x) = 2 \) and \( g'(x) = \sec^2 x \). Thus, re-evaluate the limit using this derivative information.

Step 4: Recalculating the Limit

Now, the expression under L'Hôpital's becomes \( \lim_{x \to 0} \frac{2}{\sec^2 x} = \lim_{x \to 0} 2 \cos^2 x \). Substitute \( x = 0 \) to get \( 2 \cos^2(0) = 2 \times 1^2 = 2 \).

Step 5: Conclusion and Verification

Since substituting directly after applying L'Hôpital's Rule gives a finite result and matches both the original limit's criteria and check via the derivative, the limit exists and is equal to 2.

Key Concepts

L'Hôpital's Rule

When dealing with limits that present indeterminate forms such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), L'Hôpital's Rule comes to the rescue. This rule provides a systematic way to evaluate such tricky limits. To apply L'Hôpital's Rule, first ensure that the function, when directly substituted, gives an indeterminate form. Once confirmed, you can use the formula:

• If \(\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{0}{0}\) or \(\frac{\infty}{\infty}\), then:
• \(\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}\), provided this latter limit exists.

Here, derivatives \(f'(x)\) and \(g'(x)\) pertain to the functions \(f(x)\) and \(g(x)\). Yet, it’s crucial to validate the rule’s applicability at each derivative step. Re-evaluation may be necessary if another indeterminate form arises.

Indeterminate Forms

Indeterminate forms are often encountered in calculus, especially in limits. These forms are expressions where the limits cannot be directly evaluated because substitution leads to forms like \( \frac{0}{0} \), \( \infty - \infty \), or \( 1^\infty \), to name a few.
For the expression \( \lim_{x \rightarrow 0} \frac{2x}{\tan x} \), substituting \( x = 0 \) gives \( \frac{0}{0} \), which is indeterminate.

• \( \frac{0}{0} \): Often the most common form where L'Hôpital's Rule can be applied to resolve the expression.
• Other forms may require different strategies such as trigonometric identities or series expansions for simplification.

Recognizing these forms is the first step in deciding the appropriate method to resolve the limit.

Trigonometric Limits

Trigonometric limits frequently involve expressions with sine, cosine, tangent, and other trigonometric functions. These functions have unique properties that allow for the simplification of limits, especially near zero, as they tend to produce indeterminate forms.
For instance, evaluating \( \lim _{x \rightarrow 0} \frac{2 x}{\tan x} \) involves the trigonometric limit of \( \tan x \).
The tangent function \( \tan x = \frac{\sin x}{\cos x} \) can be complex near zero but has well-known limits, e.g., \( \lim _{x \to 0} \frac{\sin x}{x} = 1 \).

• Such trigonometric properties facilitate direct substitution when the function simplifies.
• Alternatively, identities like \( \tan x = \sin x / \cos x \) are handy for rewriting expressions into solvable forms.

Utilizing these foundational limits and identities allows for the simplification and resolution of complex trigonometric limits.