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Chapter 4: Problem 69

Determine whether the limit exists, and where possible evaluate it. $$\lim _{t \rightarrow 0} \frac{\sin ^{2} A t}{\cos A t-1}, A \neq 0$$

Short Answer

Expert verified
The limit exists and equals \(-2\).

Step by step solution

01

Apply Limit Substitution

Substitute the limit value into the expression directly to check if the expression is initially defined. When we insert \( t = 0 \), we get \( \frac{\sin^2(A \cdot 0)}{\cos(A \cdot 0) - 1} = \frac{0}{1 - 1} = \frac{0}{0} \). This is an indeterminate form.
02

Use L'H么pital's Rule

Since the expression is in an indeterminate form \( \frac{0}{0} \), we can apply L'H么pital's Rule, which requires taking the derivative of the numerator and the denominator separately. The original limit now becomes \( \lim_{t \to 0} \frac{2A\sin(A t)\cos(A t)}{-A\sin(A t)} \).
03

Simplify the Expression

Simplify the fraction by canceling common factors. We notice that \( A \sin(A t) \) appears in both the numerator and the denominator. The expression simplifies to \( -2\cos(A t) \).
04

Evaluate the Simplified Limit

Substitute \( t = 0 \) into the simplified expression to find the limit: \( -2\cos(A \times 0) = -2\cos(0) = -2 \cdot 1 = -2 \).
05

Confirm the Limit Exists

Since \( -2 \) is a constant, the limit exists and is equal to \(-2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

L'H么pital's Rule
L'H么pital's Rule is a powerful method in calculus when dealing with limits that result in indeterminate forms. When you first substitute the limit value into a function and end up with expressions like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), these are known as indeterminate forms. This rule allows us to differentiate the numerator and the denominator separately and then take the limit again.

To apply the rule:
  • Ensure both the numerator and the denominator are differentiable at the point in question.
  • Differentiate both the numerator and the denominator separately.
  • Recompute the limit with these new functions.
If the new expression can be evaluated directly, it will give you the limit. If another indeterminate form appears, L'H么pital's Rule can often be applied again until a definitive value is found.
Indeterminate Forms
Indeterminate forms arise when we try to calculate a limit and end up with an unpredictable situation, such as \( \frac{0}{0} \). These forms indicate that more analysis is needed to evaluate the limit.

The most common indeterminate forms are:
  • \( \frac{0}{0} \)
  • \( \frac{\infty}{\infty} \)
  • \( \infty - \infty \)
  • \( 0 \times \infty \)
  • \( 0^0 \)
  • \( 1^\infty \)
  • \( \infty^0 \)
Each form requires a different approach, such as applying L'H么pital's Rule, simplifying the expression further, or using series expansions to resolve them into a form that allows you to find the limit.
Trigonometric Limits
Understanding trigonometric limits is essential when handling expressions involving sine and cosine functions as they approach particular values. For trigonometric limits as \( t \rightarrow 0 \), certain identities and standard limits can simplify the process.

Key strategies include:
  • Recognizing fundamental limits, such as \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \).
  • Using trigonometric identities to rewrite complex expressions into simpler ones.
  • Employing L'H么pital's Rule when indeterminate forms are present, as in the exercise \( \frac{\sin^2(At)}{\cos(At) - 1} \).
By becoming familiar with these foundational concepts, students can more easily tackle trigonometric limits without getting stuck on complex manipulations.

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Most popular questions from this chapter

(a) Find \(d^{2} y / d x^{2}\) for \(x=t^{3}+t, y=t^{2}\) (b) Is the curve concave up or down at \(t=1 ?\)

Describe the form of the limit \((0 / 0\) \(\left.\infty / \infty, \infty \cdot 0, \infty-\infty, 1^{\infty}, 0^{0}, \infty^{0}, \text { or none of these }\right) .\) Does I'Hopital's rule apply? If so, explain how. $$\lim _{x \rightarrow \infty} \frac{x}{e^{x}}$$

Let \(f(x)=x^{2} .\) Decide if the following statements are true or false. Explain your answer. \(f\) has a global minimum on any interval \([a, b]\)

Give an example of: A limit of a rational function for which I'Hopital's rule cannot be applied.

Evaluate the limit using the fact that $$\lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^{n}=e$$ $$\lim _{x \rightarrow 0^{+}}(1+x)^{1 / x}$$
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