Problem 18 - Single Variable Calculus

Question

Find the indicated limit. \(\lim _{x \rightarrow \pi / 4}(x \tan x)\)

Step-by-Step Solution

Answer

The short answer is: \(\lim _{x \rightarrow \pi / 4}(x \tan x) = \pi / 4\).

Step 1: Identify the limit notation and function

We are given the limit notation: \(\lim _{x \rightarrow \pi / 4}(x \tan x)\) Here, the function is f(x) = x * tan(x), and we need to find its limit as x approaches π/4.

Step 2: Substitute the value of x approaching into the function

To find the limit, we need to substitute the value x approaching, which is π/4, into the function: \(f(x) = x \tan x\)

Step 3: Evaluate the limit using limit properties and trigonometric identities

Using limit properties, we can rewrite the limit as: \(\lim _{x \rightarrow \pi / 4}(x \tan x) = \lim _{x \rightarrow \pi / 4}(x) \times \lim _{x \rightarrow \pi / 4}(\tan x)\) Now, let's evaluate each part separately: - The limit of x as x approaches π/4 is simply π/4, as \(x \rightarrow \pi / 4\). - The limit of tan(x) as x approaches π/4 can be found using the fact that \(\tan (\pi / 4) = 1\), because the sine and cosine of π/4 are equal to each other.

Step 4: Calculate the final limit

Now, we can find the final limit by multiplying the limits found in step 3: \(\lim _{x \rightarrow \pi / 4}(x \tan x) = \left(\pi / 4\right) \times 1\) Hence, the limit is: \(\lim _{x \rightarrow \pi / 4}(x \tan x) = \pi / 4\)

Key Concepts

Trigonometric Limits

Understanding trigonometric limits is crucial when dealing with functions involving trigonometric expressions, especially when a variable approaches a specific angle. In this exercise, we're dealing with the tangent function, \( \tan(x) \), which is a ratio of the sine and cosine functions. When finding the limit of a trigonometric function as \( x \) approaches a particular value, knowing key trigonometric identities can be extremely helpful.
It's important to remember certain special angles, such as \( \pi/4 \), \( \pi/3 \), and their associated function values to make calculations easier. For example:

• \( \tan(\pi/4) = 1 \), since the sine and cosine of \( \pi/4 \) are equal.
• Understanding these values helps in quickly determining limits involving trigonometric functions.

Recognizing these trigonometric relationships allows us to substitute directly and simplify the expression, leading to a more straightforward evaluation of the limit.

Limit Properties

Limit properties are rules that help to simplify the process of finding limits. They're like tools that we use to break down more complex expressions. One of these properties is the limit of a product, which states that the limit of the product of two functions is equal to the product of their limits, if both limits exist. This is expressed as:

• \( \lim_{x \to a} (f(x) \cdot g(x)) = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x) \)

This property was used in the original solution. It allowed us to separate \( \lim_{x \to \pi/4} (x \cdot \tan x) \) into two simpler limits: \( \lim_{x \to \pi/4} x \) and \( \lim_{x \to \pi/4} \tan x \).
The fundamental limit properties simplify the calculation process and are especially helpful in dealing with complex functions by breaking them down into more manageable pieces.

Calculus Techniques

Calculus provides various techniques for evaluating limits, which involve different approaches based on the problem's complexity. Substitution is a basic technique where you directly plug in the value into the function if it's defined. In this exercise, direct substitution works perfectly because both \( x \) and \( \tan(x) \) are well-defined at \( x = \pi/4 \).
Another technique involves using trigonometric identities or simplifications to handle undefined forms. In cases where direct substitution leads to an indeterminate form, like \( 0/0 \) or \( \infty/\infty \), more advanced techniques, such as Lopital's rule, might be necessary. But in this scenario, direct substitution suffices due to the solvability of \( \tan(\pi/4) \)=1.
Understanding when to apply different calculus techniques is essential for efficiently finding limits in a range of problems, ensuring accuracy and clarity in the solutions.