Problem 4 - Calculus Early Transcendentals: Pearson New International Edition

Question

Evaluate each limit. $$ \lim _{x \rightarrow 0} \frac{3 x \tan x}{\sin x} $$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Substitute the limit into the function

Before applying any specific limit evaluation techniques, we can substitute the value that \(x\) approaches into the function. Since this could result in an indeterminate form, we note that \(\lim _{x \rightarrow 0} \frac{3 x \tan x}{\sin x}\) can be problematic directly because substituting \(x = 0\) gives us \(\frac{0}{0}\) which is indeterminate.

Step 2: Use Trigonometric Identities

Recall the trigonometric identity for \(\tan x\), which is \(\tan x = \frac{\sin x}{\cos x}\). Substitute this into the limit function:\[\lim _{x \rightarrow 0} \frac{3x \cdot \frac{\sin x}{\cos x}}{\sin x} = \lim _{x \rightarrow 0} \frac{3x \sin x}{\sin x \cdot \cos x}.\]

Step 3: Simplify the Expression

Observe that the \(\sin x\) terms cancel each other:\[\lim _{x \rightarrow 0} \frac{3x}{\cos x}.\] Now the limit has been simplified to \(\lim _{x \rightarrow 0} \frac{3x}{\cos x}\).

Step 4: Evaluate the Limit

Since the expression is now \(\frac{3x}{\cos x}\), substitute \(x = 0\):\[\frac{3 imes 0}{\cos(0)} = \frac{0}{1} = 0.\] Thus, the limit is equal to 0.

Key Concepts

Indeterminate Forms

In calculus, an indeterminate form is a mathematical expression that arises when evaluating certain limits. These forms can originally give misleading results such as \(0/0\), \(\infty/\infty\), or others that do not suggest a definite outcome. When you substituted \(x = 0\) into \(\lim _{x \rightarrow 0} \frac{3 x \tan x}{\sin x}\), you encountered the form \(\frac{0}{0}\). This is called an indeterminate form, indicating that further analysis is necessary. The presence of an indeterminate form shows that direct substitution doesn't work, and you'll need to employ specific techniques to resolve it. These techniques include algebraic manipulation, trigonometric identities, or methods like L'Hôpital's Rule, which can help to simplify the expression and find the actual limit.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable. They help in rewriting expressions to make them easier to work with. In the case of the limit \(\lim _{x \rightarrow 0} \frac{3 x \tan x}{\sin x}\), the identity \(\tan x = \frac{\sin x}{\cos x}\) was utilized. By substituting \(\tan x\) with this identity, the expression becomes:

• \(\lim _{x \rightarrow 0} \frac{3x \cdot \frac{\sin x}{\cos x}}{\sin x}\)

This rewrite exposed factors that could be canceled, simplifying the limit problem. Recognizing and applying trigonometric identities effectively can often transform complex expressions into simpler ones, making evaluation possible. Other common identities include the Pythagorean identities, which involve \(\sin^2 x + \cos^2 x = 1\), among others, frequently used in calculus problems.

Limit Evaluation Techniques

Limits can be calculated using a variety of evaluation techniques to get past indeterminate forms or to handle complex expressions. For the limit \(\lim _{x \rightarrow 0} \frac{3 x \tan x}{\sin x}\), once it was rewritten using the trigonometric identity, re-examining the simplified form made it straightforward. The expression became a lot easier:

• \(\lim _{x \rightarrow 0} \frac{3x}{\cos x}\)

In this situation, direct substitution after simplification worked perfectly because \(\cos 0 = 1\) and the expression reduced to \(\frac{0}{1}\), which is zero. Other limit evaluation techniques include factoring, using the squeeze theorem, or L'Hôpital's Rule if you still have indeterminate forms like \(\frac{0}{0}\). Understanding when and how to apply these techniques is crucial for solving limit problems efficiently.

Simplifying Expressions

Simplifying expressions in calculus not only makes solving problems easier but can also reveal useful patterns. Simplification is often necessary when dealing with limits as shown in the exercise \(\lim _{x \rightarrow 0} \frac{3 x \tan x}{\sin x}\). By rewriting the trigonometric term in a simpler form, you eliminated the \(\sin x\) that appeared both in the numerator and denominator:

• \(\lim _{x \rightarrow 0} \frac{3x}{\cos x}\)

Here, canceling out terms drastically reduced complexity, leading to a limit that was easy to evaluate. Simplifying can involve canceling terms, factoring them, and using identities. Each step brings you closer to a clearer, often more manageable problem statement that can be directly evaluated. Always aim to simplify first to be able to see clearly how to proceed with solving limits or other calculus problems.