Problem 33 - Calculus Early Transcendentals

Question

Limits of composite functions Evaluate each limit and justify your answer. $$\lim _{x \rightarrow 0} \ln \frac{2 \sin x}{x}$$

Step-by-Step Solution

Answer

Question: Evaluate the limit and justify your answer: $$\lim_{x \rightarrow 0} \ln\frac{2 \sin x}{x}$$ Answer: The limit is $$\lim_{x \rightarrow 0} \ln\frac{2 \sin x}{x} = \ln 2$$, using L'Hôpital's Rule and limit properties of composite functions and trigonometric functions.

Step 1: Check if we can apply L'Hôpital's Rule

Let's rewrite the given limit in a form suitable to apply L'Hôpital's Rule. We know that \(ln(y)\) is continuous for \(y > 0\), so it safely can be written as: $$\lim _{x \rightarrow 0} \ln \frac{2 \sin x}{x} = \ln \lim_{x \rightarrow 0} \frac{2 \sin x}{x}$$ Now let's examine the function inside the limit: $$\lim_{x \rightarrow 0} \frac{2 \sin x}{x}$$ We notice that at x = 0, this function takes the indeterminate form \(\frac{0}{0}\). This means we can apply L'Hôpital's Rule to evaluate this limit.

Step 2: Apply L'Hôpital's Rule

L'Hôpital's Rule states that if the limit $$\lim_{x \rightarrow c} \frac{f(x)}{g(x)}$$ is \(\frac{0}{0}\) or \(\pm \frac{\infty}{\infty}\), then $$\lim_{x \rightarrow c} \frac{f(x)}{g(x)}=\lim_{x \rightarrow c} \frac{f'(x)}{g'(x)}$$ if the latter limit exists. We have \(f(x)=2\sin x\) and \(g(x)=x\). Now we compute their derivatives: $$f'(x) = 2\cos x$$ $$g'(x) = 1$$ Now we evaluate the limit as x approaches 0: $$\lim_{x \rightarrow 0} \frac{2 \cos x}{1} = \frac{2 \cos 0}{1} = 2$$

Step 3: Substitute back into the original expression

Now we have the limit of the inner function: $$\lim_{x \rightarrow 0} \frac{2 \sin x}{x} = 2$$ Now we can substitute this value back into the natural logarithm expression: $$\lim _{x \rightarrow 0} \ln \frac{2 \sin x}{x} = \ln \lim_{x \rightarrow 0} \frac{2 \sin x}{x} = \ln 2$$ Therefore, the given limit is: $$\lim_{x \rightarrow 0} \ln\frac{2 \sin x}{x} = \ln 2$$ which is the solution to the exercise.

Key Concepts

L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool in calculus used to evaluate limits that have indeterminate forms. When you encounter a limit like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), this rule can help. The rule states that:

• If you have \(\lim_{x \rightarrow c} \frac{f(x)}{g(x)}\) that results in an indeterminate form,
• Then \(\lim_{x \rightarrow c} \frac{f(x)}{g(x)} = \lim_{x \rightarrow c} \frac{f'(x)}{g'(x)}\), if this latter limit exists.

To apply L'Hôpital's Rule, you first check if the limit is indeed indeterminate. If so, you find the derivatives of the numerator and denominator separately and then compute the limit again. This often simplifies what seemed impossible or complex into a solution that is easier to calculate. L'Hôpital's Rule can be repeated if necessary until you get a determinate result.

Natural Logarithm

The natural logarithm, denoted as \(\ln(x)\) or sometimes \(\log_e(x)\), is a special type of logarithm. It is the logarithm to the base \(e\), where \(e\) is approximately 2.71828, a mathematical constant. The natural logarithm has unique properties:

• \(\ln(1) = 0\)
• \(\ln(e) = 1\)
• It grows slower than any linear function as \(x\) increases.

It is often used in calculus, differential equations, and physics due to its natural relationship with growth and decay processes. In limits, it is useful because \(\ln(x)\) is continuous for \(x > 0\), allowing us to evaluate limits inside logarithmic functions more easily.

Indeterminate Forms

Indeterminate forms are expressions that do not have a straightforward value in the evaluation of limits. Common indeterminate forms include:

• \(\frac{0}{0}\)
• \(\frac{\infty}{\infty}\)
• \(\infty - \infty\)
• \(0 \times \infty\)
• \(1^\infty\)
• \(0^0\)
• \(\infty^0\)

In the context of limits, these forms indicate that you need to use alternative methods, such as algebraic manipulation or L'Hôpital's Rule, to find a precise value. Recognizing these forms is the first step to handling them correctly, making it easier to solve complex limit problems.

Derivatives

Derivatives represent the rate of change of a function relative to its variable. They provide crucial information about the behavior of functions. In mathematical terms, the derivative of a function \(f(x)\) is denoted \(f'(x)\), and it tells us the slope of the tangent to the curve at any point \(x\).

• For example, if \(f(x) = 2\sin x\), then \(f'(x) = 2\cos x\).
• For \(g(x) = x\), the derivative is \(g'(x) = 1\).

Derivatives are foundational in calculus and are utilized for solving limits, as they can reframe complex expressions into simpler forms when using L'Hôpital's Rule. They are also employed in motion equations, optimization problems, and analysis of functions' behavior.