Problem 47 - Thomas Calculus

Question

Limits with trigonometric functions Find the limits in Exercises \(43-50 .\) $$\lim _{x \rightarrow 0} \frac{1+x+\sin x}{3 \cos x}$$

Step-by-Step Solution

Answer

The limit is \(\frac{1}{3}\).

Step 1: Analyze the limit form

First, we will analyze the form of the limit as \(x\) approaches 0. Substitute 0 into the expression:\[\frac{1 + 0 + \sin(0)}{3 \cos(0)} = \frac{1}{3}\]The expression evaluates to \(\frac{1}{3}\), indicating that the form is not indeterminate.

Step 2: Substitute directly and simplify

Since the form is not indeterminate, we can directly substitute \(x = 0\) into the expression:\[\lim _{x \rightarrow 0} \frac{1+x+\sin x}{3 \cos x} = \frac{1+0+\sin(0)}{3\cos(0)}\]Simplify this to:\[\frac{1 + 0 + 0}{3 \times 1} = \frac{1}{3}\]

Step 3: Verify using trigonometric limit properties

For verification, consider the known limits:1. \(\lim_{x \to 0} \sin x = 0\)2. \(\lim_{x \to 0} \cos x = 1\)Based on these, substitute back to ensure correctness:\[\lim _{x \rightarrow 0} \frac{1+x+\sin x}{3 \cos x} = \frac{1+0+0}{3\times1} = \frac{1}{3}\]This verifies our previous calculation.

Key Concepts

Trigonometric Limits

Trigonometric limits are a fundamental part of calculus, especially when dealing with functions involving sine or cosine. When we talk about the limits of trigonometric functions, we're usually interested in how these functions behave as their inputs approach a particular value. One of the most common scenarios is when the input approaches zero.

• Sine Limit: The limit of \( \sin(x) \) as \( x \to 0 \) equals zero, which is written as \( \lim_{x \to 0} \sin(x) = 0 \).
• Cosine Limit: As \( x \) approaches zero, the cosine function approaches one: \( \lim_{x \to 0} \cos(x) = 1 \).

These characteristic limits help us understand how trigonometric functions behave near zero. They are used to simplify expressions and make the calculations of limits straightforward. Knowing these properties allows us to quickly resolve limits involving trigonometric functions without unnecessary complication.

Evaluation of Limits

Evaluating limits is a method used to find the value that a function approaches as the input approaches a given number. It involves checking the behavior of function expressions as the variable nears the value of interest. The function can be simplified or transformed for easy evaluation, especially with trigonometric functions where immediate substitution could simplify matters.
When evaluating limits like \( \lim_{x \to 0} \frac{1+x+\sin x}{3 \cos x} \), it is essential first to understand the form of the function as \( x \) approaches 0. If the form produces a finite value, only then can we substitute the value directly.

• If substitution leads to a form like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), these are indeterminate forms and require special techniques like L'Hôpital's Rule.
• However, if direct substitution results in a clear value, as seen in the problem example, the limit can be computed easily without any further complication.

In our particular exercise, substituting \( x = 0 \) gives us a straightforward value \( \frac{1}{3} \), illustrating that not all limits necessitate complex evaluation.

Non-Indeterminate Forms

Non-indeterminate forms in limits occur when the limit expression directly resolves to a specific value without yielding an ambiguous result like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). In these cases, straightforward substitution of the value can be employed to evaluate the limit.
Consider the expression \( \lim_{x \to 0} \frac{1+x+\sin x}{3 \cos x} \); when \( x = 0 \) is substituted, the expression resolves immediately to \( \frac{1}{3} \). This smooth transition from expression to value indicates a non-indeterminate form, where the substitution results in a finite conclusion without further manipulation.

• Such limits demonstrate the stability and predictability of some functions at specific points.
• Mathematicians often appreciate these forms because they provide clear insights without needing complex calculus techniques.

Recognizing non-indeterminate forms is crucial as it allows one to quickly ascertain the limit's result, streamlining the problem-solving process in calculus.