Problem 40 - University Calculus: Early Transcendentals

Question

Use I'Hópital's rule to find the limits. $$\lim _{x \rightarrow 0^{+}}\left(\frac{3 x+1}{x}-\frac{1}{\sin x}\right)$$

Step-by-Step Solution

Answer

The limit is 3.

Step 1: Simplify the Expression

First, find a common denominator for the two fractions in the expression to combine them. \[\frac{3x + 1}{x} - \frac{1}{\sin x} = \frac{(3x + 1)\sin x - x}{x \sin x}\]Now, simplify the numerator.\[(3x + 1)\sin x - x = 3x \sin x + \sin x - x\]

Step 2: Evaluate the Form of the Limit

Recognize that as \(x\) approaches 0 from the positive side, both the simplified numerator and denominator approach 0. This suggests an indeterminate form \(\frac{0}{0}\), making it suitable for L'Hôpital's Rule.

Step 3: Apply L'Hôpital's Rule

Compute the derivatives of the numerator and the denominator.The numerator:\[\frac{d}{dx}(3x \sin x + \sin x - x) = 3\sin x + 3x \cos x + \cos x - 1\]The denominator:\[\frac{d}{dx}(x \sin x) = \sin x + x \cos x\]Apply L'Hôpital's Rule:\[\lim_{x \to 0^+} \frac{3\sin x + 3x \cos x + \cos x - 1}{\sin x + x \cos x}\]

Step 4: Simplify After Applying L'Hôpital's Rule

Evaluate the limit expression again at \(x \rightarrow 0^{+}\).Substitute \(x = 0\) in all trigonometric expressions:\[\sin(0) = 0 \quad \text{and} \quad \cos(0) = 1\]Thus, the expression becomes:\[\frac{3(0) + 3(0)(1) + 1 - 1}{0 + 0} = \frac{0}{0}\]Since it's still an indeterminate form, apply L'Hôpital's Rule again.Compute the second derivatives:Numerator:\[\frac{d}{dx}(3\sin x + 3x \cos x + \cos x - 1) = 3\cos x + 3\cos x - 3x \sin x - \sin x\]Denominator:\[\frac{d}{dx}(\sin x + x \cos x) = \cos x + \cos x - x \sin x\]Re-evaluate limit:\[\lim_{x \to 0^+} \frac{6\cos x - \sin x - 3x \sin x}{2\cos x - x \sin x} = \frac{6}{2} = 3\]

Step 5: Conclude the Result

Thus, the limit of the original expression as \(x\) approaches 0 from the positive side is 3.

Key Concepts

Calculus Limits

Calculus limits are an essential part of understanding advanced mathematical concepts. They allow us to understand the behavior of functions as they approach a specific point or value. At its core, a limit describes what happens to a function when the input (or variable) approaches a particular value. When we say that the limit of a function as the variable approaches a certain value is a number, it means that the function gets very close to that number as the variable gets closer to the specified value.There are several ways limits can be expressed:

• Finite limits at a point: When the function approaches a specific value as the input approaches a number.
• Limits at infinity: When the input grows indefinitely large, and we are interested in the behavior of the function.
• One-sided limits: These focus on behavior from just one direction (either from the left or right of the point).

In problems where limits result in an undefined form like \(\frac{0}{0}\), techniques like L'Hôpital's Rule help in further analysis.

Indeterminate Forms

Indeterminate forms arise when evaluating a limit results in an expression that doesn't directly imply a specific number. The most common form is \(\frac{0}{0}\), but others include \infty - \infty\ and \frac{\infty}{\infty}\.These forms indicate that more work is needed to find the limit. For example, L'Hôpital's Rule is a valuable technique to resolve such forms by taking derivatives. Here's how it works:

• Identify the indeterminate form, like \(\frac{0}{0}\).
• Take the derivative of the numerator and the denominator separately.
• Then, calculate the limit of the resulting expression.

In the original exercise, the limit initially produced \(\frac{0}{0}\), which suggested using L'Hôpital's Rule. By applying it, the complex trigonometric expression simplifies significantly, helping find the correct behavior of the function at that point.

Trigonometric Limits

Understanding limits involving trigonometric functions is crucial since these often appear in Calculus problems. Trigonometric limits can include expressions involving \(\sin(x)\), \(\cos(x)\), and other trigonometric functions as they approach a particular angle.A few noteworthy limits to remember are:

• The limit of \(\frac{\sin x}{x}\) as \(x\) approaches 0 is 1. This helps in simplifying problems involving small angle approximations.
• The limit of \(\cos x\) as \(x\) approaches 0 is straightforwardly 1.

In the original problem, applying these trigonometric limit properties at \(x = 0\) simplified the task. Recognizing that \(\sin(0) = 0\) and \(\cos(0) = 1\) allowed direct substitution, which made evaluating the limit more manageable after applying L'Hôpital's Rule. These fundamental identities for trigonometric functions form the basis for tackling a wide array of limit problems.