Problem 57 - University Calculus: Early Transcendentals

Question

Is there a value of \(c\) that will make $$f(x)=\left\\{\begin{array}{ll} \frac{\sin ^{2} 3 x}{x^{2}}, & x \neq 0 \\ c, & x=0 \end{array}\right.$$ continuous at \(x=0 ?\) Give reasons for your answer.

Step-by-Step Solution

Answer

Yes, setting \(c = 9\) makes \(f(x)\) continuous at \(x=0\).

Step 1: Understanding Continuity

For a function to be continuous at a point, the limit of the function as it approaches the point from both sides must equal the function’s value at that point. For this exercise, to determine if \(f(x)\) is continuous at \(x = 0\), we need to find the limit as \(x\) approaches 0.

Step 2: Substituting for Limit

To evaluate \( \lim_{{x \to 0}} \frac{\sin^{2} 3x}{x^{2}} \), let's substitute \(y = 3x\), then \( x = \frac{y}{3} \) and when \(x \to 0\), \(y \to 0\). The expression becomes \( \lim_{{y \to 0}} \frac{\sin^{2} y}{\left(\frac{y}{3}\right)^{2}} = \lim_{{y \to 0}} \frac{9 \sin^{2} y}{y^{2}} \).

Step 3: Applying L'Hôpital's Rule

This limit is of the form \(\frac{0}{0}\), allowing the use of L'Hôpital's Rule. By applying it, differentiate the numerator and the denominator: \[ \lim_{{y \to 0}} \frac{9 \sin^{2} y}{y^{2}} = \lim_{{y \to 0}} \frac{18 \sin y \cos y}{2y} \].

Step 4: Applying L'Hôpital's Rule Again

The expression remains \( \frac{0}{0} \), so apply L'Hôpital's Rule once more. Differentiate again: \[ \lim_{{y \to 0}} \frac{18 \sin y \cos y}{2y} = \lim_{{y \to 0}} \frac{18(\cos^2 y - \sin^2 y)}{2} \].

Step 5: Final Limit Calculation

At \(y = 0\), \( \sin y \to 0\) and \( \cos y \to 1\), thus the limit simplifies to \( \lim_{{y \to 0}} \frac{18(1 - 0)}{2} = \frac{18}{2} = 9 \).

Step 6: Choosing the Correct Value of \(c\)

For \(f(x)\) to be continuous at \(x = 0\), the value of \(f(0) = c\) must equal the limit we found as \(x\) approaches 0. Therefore, set \(c = 9\).

Key Concepts

L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool used in calculus to evaluate limits that result in indeterminate forms like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). When evaluating such limits, this rule allows us to differentiate the numerator and the denominator separately, then take the limit again.
Let's break the process down a bit:

• Identify the limit's form: Start by checking if the given limit is in one of the indeterminate forms.
• Differentiate numerator and denominator: Apply the rule by taking the derivative of the numerator and the derivative of the denominator separately.
• Re-evaluate the limit: After differentiation, substitute the value into the limit again.
• If needed, apply the rule more than once: Sometimes, the form after the first application is still indeterminate, so you may apply L'Hôpital's Rule again.

In the original solution, L'Hôpital's Rule was applied twice to the limit \(\lim_{{y \to 0}} \frac{9 \sin^{2} y}{y^{2}}\) to finally determine the limit as 9. This illustrates the flexibility and resilience of the rule in dealing with complex limits.

Trigonometric Limits

Trigonometric limits often occur in calculus, especially with functions involving sine or cosine near points of discontinuity or zero. Understanding these limits can be crucial when proving continuity or differentiating functions.
Key points to grasp:

• Sine and cosine functions are bounded: \(-1 \leq \sin \theta \leq 1\) and \(-1 \leq \cos \theta \leq 1\).
• Important limits: \(\lim_{x \to 0} \frac{\sin x}{x} = 1\) and \(\lim_{x \to 0} \frac{1 - \cos x}{x} = 0\).
• Scaling factor: If \(\sin kx\) or \(\cos kx\) appear, consider scaling \(x\) to simplify the calculations.

In the exercise, setting \(y = 3x\) was a technique used to handle the \(\sin^{2} 3x\) term more effectively and easily manage trigonometric identities when applying limits. Mastery of such conversions facilitates smoother evaluations of trigonometric limits.

Piecewise Functions

Piecewise functions are defined with different expressions over various intervals. These are particularly common in real-world applications where a situation can change based on conditions or inputs.
To analyze continuity in piecewise functions:

• Check each piece's domain: Ensure that each function segment is defined over its respective interval.
• Evaluate limits at transition points: Wherever the function definition changes from one expression to another, compute the limit from both sides.
• Match function values at endpoints: Check that the piecewise function's value at each transition equals the limits from both sides.

In the original problem, \(f(x)\) needed to be continuous at \(x = 0\). For this, the limit of the expression as \(x\) approached 0 needed to equal the function value exactly at \(x = 0\). This necessitated setting the constant \(c\) in the piecewise definition to match the calculated limit, which was determined using L'Hôpital's Rule.