Problem 48 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \lim _{x \rightarrow \frac{\pi}{2}} \frac{1}{1+e^{\tan x}}\\{\text { Ans. } 1,0\\} $$

Step-by-Step Solution

Answer

The short answer to the given limit problem is: \(\lim _{x \rightarrow \frac{\pi}{2}} \frac{1}{1+e^{\tan x}} = 0\).

Step 1: Analyze the given function

Let's analyze the given function: \[ f(x) = \frac{1}{1+e^{\tan x}} \] As x approaches π/2, the value of tan(x) approaches infinity. Therefore, we should analyze the behavior of the denominator term as tan(x) approaches infinity.

Step 2: Analyze the behavior of the denominator

As tan(x) approaches infinity: \[ 1 + e^{\tan x} \rightarrow 1 + e^{\infty} \rightarrow \infty \] Now we can conclude that as x approaches π/2, thee denominator approaches infinity, and the function approaches 0. Hence, the limit exists, and we can evaluate the limit.

Step 3: Evaluate the limit

Since we've determined that f(x) approaches 0 as the denominator goes to infinity, we can now evaluate the limit as follows: \[ \lim _{x \rightarrow \frac{\pi}{2}} \frac{1}{1+e^{\tan x}} = 0 \] So, the answer to the limit is 0.

Key Concepts

Trigonometric limits

Trigonometric limits involve finding the behavior of functions with trigonometric terms as the input approaches a specific value. When analyzing limits that involve trigonometric functions, such as sine, cosine, and tangent, we often look at specific angles due to the periodic nature of these functions.
For instance, the tangent function, \(^\tan(x)\), can present challenges because as \(^x\) approaches certain critical angles like \(^\frac{\pi}{2}\), the value of \(^\tan(x)\) tends towards infinity. This makes it crucial to consider asymptotes, where the function value becomes infinitely large, as seen in this exercise. Understanding these behaviors can simplify the process of evaluating complicated trigonometric limits.

Infinity limits

Infinity limits deal with the behavior of functions as the input approaches infinity or negative infinity. These types of limits often help determine asymptotic behavior and end behavior of functions.

• If a function grows indefinitely as the input grows, we say the function approaches infinity.
• Conversely, if the function decreases without bound, it approaches negative infinity.

The exercise showcases \(^\tan(x)\), which tends to infinity as \(^x\) approaches \(^\frac{\pi}{2}\). Recognizing this helps in analyzing how the function behaves in the vicinity of vertical asymptotes, leading to clearer insights about the problem at hand.

Exponential functions

Exponential functions, characterized by expressions like \(^e^{\tan(x)}\), play a significant role in evaluating limits. With base \(^e\), these functions grow extremely rapidly.

• When the exponent is positive and large, \(^e^x\) increases without bound.
• Conversely, if negative and large in absolute value, the term diminishes towards zero.

This rapid growth is instrumental in determining limits involving exponential terms, particularly when the input, such as the angle in a tangent function, approaches values that cause the exponent to become significantly large.

Evaluating limits step-by-step

Evaluating limits step-by-step involves breaking down the function into manageable parts to analyze its behavior as the input approaches a certain value. This methodical approach helps simplify complex problems.

• First, identify what happens to each component of the function as the variable approaches the point of interest.
• Then, determine overall behavior based on these findings.

For example, in the original solution, analyzing \(^1 + e^{\tan x}\) provides insights about dominator behavior, leading to conclusions about the overall function limit. Breaking it down into steps aids in constructing a clear understanding, fostering better problem-solving skills.