Problem 34 - Calculus - AP Edition

Question

Find the limits. $$\lim _{x \rightarrow 0}\left(\frac{1}{x^{2}}-\frac{\cos 3 x}{x^{2}}\right)$$

Step-by-Step Solution

Answer

The limit is \( \frac{9}{2} \).

Step 1: Identify the Expression

We need to find the limit of the expression \( \lim _{x \rightarrow 0} \left( \frac{1}{x^{2}}-\frac{\cos 3 x}{x^{2}} \right) \). The expression can be rewritten as \( \lim_{x \rightarrow 0} \frac{1 - \cos 3x}{x^2} \).

Step 2: Use Trigonometric Identity

Recall the trigonometric identity for small angles: \(1 - \cos \theta \approx \frac{\theta^2}{2}\) when \(\theta\) is near zero. We use this to approximate \(1 - \cos 3x\).

Step 3: Substitute the Approximation

Substitute \(1 - \cos 3x \approx \frac{(3x)^2}{2} = \frac{9x^2}{2}\) into the expression, yielding \( \lim_{x \rightarrow 0} \frac{\frac{9x^2}{2}}{x^2} \).

Step 4: Simplify the Expression

Simplify \( \frac{\frac{9x^2}{2}}{x^2} = \frac{9}{2} \). This simplifies the limit to \( \lim_{x \rightarrow 0} \frac{9}{2} \).

Step 5: Evaluate the Limit

Since there are no \(x\) variables remaining in the expression after simplification, the limit is simply the constant value of the simplified expression.

Key Concepts

Trigonometric Limits

Trigonometric limits are often encountered when dealing with expressions involving trigonometric functions like sine, cosine, and tangent as they approach certain angles. A key property of trigonometric limits is their behavior near specific points, such as zero. In the given problem, we deal with the limit of \(\lim_{x \rightarrow 0} \left( \frac{1 - \cos 3x}{x^2} \right)\). As it approaches zero, it is crucial to use known trigonometric identities to simplify and solve the limit.
For cosine, especially as it approaches 0, an essential identity is \(1 - \cos \theta \), which can be approximated using small angle approximations. This section sets the groundwork for understanding how trigonometric identities simplify limits.

Small Angle Approximation

When dealing with limits that include trigonometric functions like cosine, especially when the angle \(\theta\) approaches zero, we use small angle approximations. For small angles, we have the approximation \(1 - \cos \theta \approx \frac{\theta^2}{2}\). This formula is extremely useful because it reduces complex trigonometric functions into simpler polynomial expressions.
In our exercise, \({\theta = 3x}\) turns into \(1 - \cos 3x \approx \frac{(3x)^2}{2} = \frac{9x^2}{2}\). Here, \(3x\) is considered a small angle when \(x\) tends to zero. Using this approximation allows for a straightforward substitution back into the limit expression, facilitating further simplification.

Limit Simplification

Having performed the small angle approximation, the next step involves simplifying the limit expression. The original complex expression \(\lim_{x \rightarrow 0} \frac{1 - \cos 3x}{x^2}\) is transformed using the simplified form \(\frac{9x^2}{2}\).
The expression becomes \(\lim_{x \rightarrow 0} \frac{\frac{9x^2}{2}}{x^2}\). This can be simplified by canceling out the \(x^2\) terms, leaving us with \(\frac{9}{2}\).
The remaining expression, \(\lim_{x \rightarrow 0} \frac{9}{2}\), does not contain \(x\) and thus simplifies directly to the constant \(\frac{9}{2}\), the evaluated limit. This process exemplifies how applying basic algebraic manipulations and simplifications help solve seemingly complicated limit problems.