Problem 17 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The function \(f(x)=\operatorname{are} \tan \frac{1}{x-5}\) has (A) discontinuity of the first kind at \(x=5\) (B) discontinuity of the second kind at \(x=5\) (C) removable discontinuity at \(x=5\) (D) continuous at \(x=5\)

Step-by-Step Solution

Answer

The function has a discontinuity of the second kind at \( x = 5 \).

Step 1: Identify the Function Type

The function given is \( f(x) = \tan\left(\frac{1}{x-5}\right) \). This is a trigonometric function involving tangent, which may have discontinuities at points where its argument approaches integer multiples of \( \frac{\pi}{2} \), leading to division by zero situations in the tangent function.

Step 2: Analyze the Argument of the Function

The argument of the tangent function is \( \frac{1}{x-5} \). As \( x \) approaches 5, the value of \( \frac{1}{x-5} \) approaches infinity or negative infinity, depending on the direction of approach, thus causing the tangent function to oscillate between negative and positive infinity.

Step 3: Determine the Type of Discontinuity

Since \( \tan\left(\frac{1}{x-5}\right) \) oscillates infinitely near \( x = 5 \), this behavior results in a discontinuity of the second kind (or essential discontinuity). This is because the function does not have a limit as \( x \) approaches 5.

Key Concepts

Trigonometric Functions

Trigonometric functions, like sine, cosine, and tangent, play a crucial role in mathematics, especially when studying angles and periodic phenomena. These functions are defined using the unit circle and exhibit unique oscillatory behavior. The tangent function, \(\tan(x)\), in particular, has points where it is not defined.
These points occur at \(x = \frac{\pi}{2} + n\pi\) for any integer \(n\), where the function approaches infinity either positively or negatively. This behavior is important when determining the continuity of functions that involve tangent as their central operation.

• The tangent function often leads to vertical asymptotes at these undefined points, with values flipping between negative and positive infinity.
• When analyzing functions like \(f(x) = \tan\left(\frac{1}{x-5}\right)\), it is critical to consider how the inner argument affects the tangent's behavior.

By understanding these characteristics, you can determine and predict where a trigonometric function might encounter difficulties or discontinuities.

Essential Discontinuity

An essential discontinuity, sometimes referred to as a discontinuity of the second kind, occurs when a function behaves erratically around a certain point with no finite limit. For the function \(f(x) = \tan\left(\frac{1}{x-5}\right)\), the essential discontinuity is found at \(x = 5\).
This is because as \(x\) gets closer to 5, the function's argument \(\frac{1}{x-5}\) tends toward infinitely large or small values, making \(\tan\left(\frac{1}{x-5}\right)\) oscillate.

• Unlike removable discontinuities, which can be 'fixed' by redefining the function at a point, essential discontinuities are inherently unstable.
• The function does not settle at a particular value, making the discontinuity non-removable.

Understanding essential discontinuities is vital when studying limits, helping to identify when and why a function cannot be made continuous at a given point.

Limits and Continuity

Limits and continuity are fundamental concepts in calculus and analysis. They allow us to understand the behavior of functions as they approach particular points or extend towards infinity. The limit of a function exists if the function approaches a specific value as the input gets closer to some point. This idea can determine whether a function is continuous or discontinuous at that point.
For the function \(f(x) = \tan\left(\frac{1}{x-5}\right)\), as \(x\) approaches 5, the results become unstable, indicating a lack of continuity.

• A continuous function means you can draw it on a graph without lifting your pencil.
• A discontinuous function, like \(f(x)\) at \(x = 5\), exhibits breaks or holes.

Recognizing the conditions under which limits do not exist is important for analyzing such functions. It is especially critical for understanding interruptions known as essential discontinuities, where no limiting value can be established.