Problem 18 - Calculus Early Transcendentals

Question

Evaluate the following limits using \(l\) Hópital's Rule. $$\lim _{x \rightarrow 1} \frac{4 \tan ^{-1} x-\pi}{x-1}$$

Step-by-Step Solution

Answer

Question: Calculate the limit of the given function as x approaches 1: $$\lim _{x \rightarrow 1} \frac{4 \tan ^{-1} x-\pi}{x-1}$$ Answer: The limit of the given function as x approaches 1 is 2.

Step 1: Check the conditions for applying L'Hôpital's Rule

As x approaches 1, the numerator function approaches \(\pi\), and the denominator approaches 0. Therefore, we cannot directly apply L'Hôpital's Rule. We will first rewrite the original limit: $$\lim _{x \rightarrow 1} \frac{4 \tan ^{-1} x-\pi}{x-1} = \lim _{h \rightarrow 0} \frac{4 \tan ^{-1} (1 + h)-\pi}{h}$$ Now, both the numerator and denominator approach 0 as h approaches 0. We can now apply L'Hôpital's Rule.

Step 2: Find the derivatives of the numerator and denominator functions

Now, we'll find the derivative of the numerator and denominator functions with respect to h. Let's denote \(f(h) = 4 \tan^{-1}(1 + h) - \pi\) and \(g(h) = h\). We have: $$f'(h) = \frac{d}{dh} (4 \tan^{-1}(1 + h) - \pi) = 4 \cdot \frac{1}{1+(1+h)^2}$$ $$g'(h) = \frac{d}{dh}(h) = 1$$

Step 3: Apply L'Hôpital's Rule and Evaluate the Limit

Now, we'll apply L'Hôpital's Rule in the rewritten limit: $$\lim _{h \rightarrow 0} \frac{4 \tan ^{-1} (1 + h)-\pi}{h} = \lim _{h \rightarrow 0} \frac{f'(h)}{g'(h)} = \lim _{h \rightarrow 0} \frac{4 \cdot \frac{1}{1+(1+h)^2}}{1}$$ To evaluate this limit, substitute h = 0 in the expression: $$\lim _{h \rightarrow 0} \frac{4 \cdot \frac{1}{1+(1+h)^2}}{1} = \frac{4 \cdot \frac{1}{1+(1+0)^2}}{1} = \frac{4 \cdot \frac{1}{1+1^2}}{1} = \frac{4}{2} = 2$$ Therefore, the limit of the given function is 2: $$\lim _{x \rightarrow 1} \frac{4 \tan ^{-1} x-\pi}{x-1} = 2$$

Key Concepts

Limits

In calculus, a limit is a fundamental concept that describes the behavior of a function as its argument approaches a certain point, either from the left, the right, or both. It is a way to express the value that a function f(x) gets closer to as x approaches a specific value. Limits are used to define many important concepts in calculus, such as continuity, derivatives, and integrals. When we write
\[ \lim_{x \to a} f(x) = L \]
we are stating that as x gets arbitrarily close to a, f(x) approaches the value L. Limits can be evaluated by substitution if the function is continuous at the point of interest. However, when it's not possible due to an undefined expression, like 0/0 or ∞/∞, other techniques, such as L'Hôpital's Rule, factoring, or rationalizing, can be employed to evaluate the limit.

Derivative of Inverse Trigonometric Functions

The derivatives of inverse trigonometric functions are particularly important when we are dealing with limits involving these functions.
For example, the derivative of the inverse tangent function, also known as arctan, is defined as:
\[ \frac{d}{dx}(\tan^{-1}(x)) = \frac{1}{1+x^2}. \]
Understanding how to compute these derivatives is essential when using L'Hôpital's Rule since this rule involves taking the derivative of functions present in the limit. In the provided example, the derivative of 4\tan^{-1}(x) when rewritten in terms of h becomes an application of the chain rule, resulting in:
\[ f'(h) = \frac{d}{dh} (4 \tan^{-1}(1 + h)) = 4 \cdot \frac{1}{1+(1+h)^2}. \]
This derivative is then used in L'Hôpital's Rule to evaluate the limit as h approaches zero.

Calculus

Calculus is a branch of mathematics focusing on rates of change (differential calculus) and accumulation of quantities (integral calculus). It deals with properties and applications of derivatives and integrals, and it plays a vital role in science, economics, and engineering.

The principles of calculus are leveraged to solve problems related to motion, area, volume, and other concepts that arise in modeling change and that require precise definitions of infinitely small quantities. The example concerning L'Hôpital's Rule showcases differential calculus at work—it involves taking derivatives and requires an understanding of how the function's behavior changes at specific points.

Evaluating Limits

Evaluating limits, especially those that present an indeterminate form such as 0/0 or ∞/∞, can be challenging. In such situations, special techniques like L'Hôpital's Rule come in handy. According to L'Hôpital's Rule, if both the numerator and denominator of a fraction tend to zero or infinity and their derivatives exist and are continuous, then
\[ \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \]
provided the limit on the right exists or is ±∞. This powerful tool in calculus is applied after verifying the conditions are met, like in the exercise we're discussing.

Here, after recognizing an indeterminate form, we found the derivatives of the functions involved and applied L'Hôpital’s Rule to successfully evaluate the limit. The result gave us a clear value of 2, confirming the limit of the function as x approaches 1.