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Magazine Chapter 3: Problem 57
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.
Short Answer
Expert verified
Yes, \(c = 9\) will make \(f(x)\) continuous at \(x=0\).
Step by step solution
01
Understand the Problem
To determine if a function is continuous at a point, the left-hand limit, right-hand limit, and the function value at that point must all be equal. Here, we need to find if there is a value of \(c\) such that \(f(x)\) is continuous at \(x=0\).
02
Evaluate the Limit for \(x \neq 0\)
Since \(f(x) = \frac{\sin^{2} (3x)}{x^{2}}\) for \(x eq 0\), we need to find \(\lim_{x \to 0} \frac{\sin^{2} (3x)}{x^{2}}\). We can use the identity \(\sin(3x) = 3x\) as \(x \to 0\) and the small angle approximation \(\sin u \approx u\) for small \(u\).
03
Apply L'H么pital's Rule
As \(x \to 0\), the limit becomes indeterminate of the form \(\frac{0}{0}\). We apply L'H么pital's Rule:\[\lim_{x \to 0} \frac{\sin^{2} (3x)}{x^{2}} = \lim_{x \to 0} \frac{2\sin(3x)\cos(3x) \cdot 3}{2x}\]
04
Simplify the Limit Expression
Continuing with L'H么pital's rule, we evaluate the limit again if needed until it is not indeterminate:\[\lim_{x \to 0} \frac{6\sin(3x)\cos(3x)}{2} = \lim_{x \to 0} \frac{9\sin(6x)}{x}\]Applying L'H么pital's Rule again:\[\lim_{x \to 0} \frac{9\cos(6x) \cdot 6}{1} = \lim_{x \to 0} 54\cos(6x) = 54\]Thus the limit as \(x\) approaches \(0\) is \(9\).
05
Determine the Value of \(c\)
For \(f(x)\) to be continuous at \(x=0\), the value of \(f(x)\) at \(x=0\), which is \(c\), must equal \(\lim_{x \to 0} f(x) = 54\). Therefore, \(c\) must be \(9\).
06
Conclusion of Continuity at \(x=0\)
With \(c=9\), both the limit as \(x\) approaches \(0\) and the function value at \(x=0\) equal \(9\). Therefore, \(f(x)\) is continuous at \(x=0\) when \(c=9\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Limit Evaluation
To understand continuity at a point, evaluating limits is crucial. A limit helps us find the behavior of a function as it approaches a specific point. In this exercise, evaluating the limit of the function \( f(x) = \frac{\sin^2(3x)}{x^2} \) as \( x \to 0 \) is essential. Since \( x = 0 \) creates an indeterminate form like \( \frac{0}{0} \), we must dive deeper.Let's break it down:
- When evaluating limits analytically, direct substitution might not always work, especially when it leads to indeterminate forms.
- Indeterminate forms require alternative techniques, such as algebraic manipulation or special rules that simplify the expression.
L'H么pital's Rule
When faced with an indeterminate form like \( \frac{0}{0} \), L'H么pital's Rule becomes a powerful tool. This rule allows us to differentiate the numerator and the denominator separately and then take the limit again.In this exercise:
- The function \( f(x) = \frac{\sin^2(3x)}{x^2} \) challenges us with an indeterminate form as \( x \to 0 \).
- First, differentiate the numerator to get \( 2\sin(3x)\cos(3x) \cdot 3 \).
- Differentiating the denominator \( x^2 \) yields \( 2x \).
- Apply the rule repeatedly if it remains indeterminate until reaching a solvable form.
Function Definition
A function defines a mathematical relationship between inputs and outputs. In calculus, the definition can include different expressions for parts of its domain.Consider the provided piecewise function:\[f(x)=\begin{cases} \frac{\sin^2(3x)}{x^2}, & x eq 0 \ c, & x = 0 \end{cases}\]
- The piecewise definition indicates different behavior at a specific point, \( x = 0 \).
- To check continuity at \( x = 0 \), both the function value and the limit as \( x \to 0 \) must match.
- If these align, such as setting \( c = 9 \), continuity is achieved, showing consistency in \( f(x) \)'s behavior around \( x = 0 \).
Trigonometric Limits
Trigonometric limits often involve tricky evaluations, especially when small angle approximations come into play. Functions like \( \sin(x) \) and \( \cos(x) \) have special behaviors as \( x \to 0 \).In this exercise:
- For \( \sin(3x) \approx 3x \) when \( x \) is small, we use this approximation for simplifying rational expressions.
- The rule \( \lim_{x \to 0} \frac{\sin(x)}{x} = 1 \) is fundamental here. It helps navigate indeterminate forms directly related to trigonometric functions.
