Problem 102 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(f\) be a function defined on an interval \(\mathrm{I}\). If \(f\) be discontinuous at a point \(p \in I\), then we say that (i) \(f\) has a removable discontinuity at \(p\) if \(\lim _{x \rightarrow p} f(x)\) exists but is not equal to \(f(p)\). (ii) \(f\) has a discontinuity of the first kind at \(p\) if \(\lim _{x \rightarrow p-0} f(x)\) and \(\lim _{x \rightarrow p+0} f(x)\) exist but are unequal. (iii) \(f\) has a discontinuity of the second kind at \(p\) if neither of \(\lim _{x \rightarrow p-0} f(x)\) and \(\lim _{x \rightarrow p+0} f(x)\) exists. 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

Choice (A): Discontinuity of the first kind at \( x=5 \).

Step 1: Analyze the Function

The function given is \( f(x) = \tan^{-1}\left(\frac{1}{x-5}\right) \). Notice that it will be undefined at \( x = 5 \) because the expression inside the \( \tan^{-1} \) function will become \( \frac{1}{0} \), which is undefined.

Step 2: Determine Limits from the Left

Examine the left-hand limit \( \lim_{x \to 5^-} f(x) \). As \( x \to 5^- \), \( \frac{1}{x-5} \to -\infty \). Thus, \( \tan^{-1}(-\infty) = -\frac{\pi}{2} \). Therefore, \( \lim_{x \to 5^-} \tan^{-1}\left(\frac{1}{x-5}\right) = -\frac{\pi}{2} \).

Step 3: Determine Limits from the Right

Examine the right-hand limit \( \lim_{x \to 5^+} f(x) \). As \( x \to 5^+ \), \( \frac{1}{x-5} \to +\infty \). Thus, \( \tan^{-1}(+\infty) = \frac{\pi}{2} \). Therefore, \( \lim_{x \to 5^+} \tan^{-1}\left(\frac{1}{x-5}\right) = \frac{\pi}{2} \).

Step 4: Compare Left and Right Limits

We found that \( \lim_{x \to 5^-} f(x) = -\frac{\pi}{2} \) and \( \lim_{x \to 5^+} f(x) = \frac{\pi}{2} \). Since they are not equal, \( \lim_{x \to 5} f(x) \) does not exist.

Step 5: Classify the Discontinuity

Since neither one-sided limit leads to \( \lim_{x \to 5} f(x) \) existing and the one-sided limits themselves are finite and exist but are unequal, \( f(x) \) has a discontinuity of the first kind.

Key Concepts

Removable Discontinuity

In calculus, a removable discontinuity occurs at a point in a function where the limit exists as you approach from either side, but the actual function value at that point does not match the limit. It's like a tiny glitch that can be "fixed" by redefining the function at that point to equal the existing limit.
\[ \lim_{x \to p} f(x) = L, \text{ but } f(p) eq L \]
In essence, the graph of the function may have a hole at this point. This type of discontinuity is often seen in situations where a function has been improperly defined, such as a rational function with a factor canceling out in both the numerator and denominator.
An example might be \( f(x) = \frac{x^2 - 1}{x - 1} \) at \( x = 1 \), where the function simplifies to \( f(x) = x + 1 \) for \( x eq 1 \), suggesting we redefine \( f(1) = 2 \) to "remove" the discontinuity.

• Limits from both sides exist and are equal
• Function value doesn't match the limit
• Fixable by adjusting the function's definition at that point

First Kind Discontinuity

A discontinuity of the first kind, also known as a jump discontinuity, is observed when the function approaches two different values from the left and right of a specific point. This means that while both one-sided limits exist, they are not equal. The graph of the function looks like a sudden jump at this point.
\[ \lim_{x \to p^-} f(x) eq \lim_{x \to p^+} f(x) \]
In real-world terms, imagine driving a car on a straight road, and suddenly, you have to jump from one height to another without any incline. This can happen in piecewise functions where different rules apply for different segments of the domain.
For instance, consider \( f(x) = \begin{cases} 2x & \text{if } x < 1 \3x & \text{if } x \geq 1 \end{cases} \). The function jumps from one function rule to another at \( x = 1 \).

• Both one-sided limits exist but are not equal
• Results in a visible "jump" in the graph
• Common in piecewise functions

Second Kind Discontinuity

Discontinuities of the second kind occur when neither of the one-sided limits at a point exists. This can result from the function approaching infinity or oscillating wildly as it approaches the point from either side.
\[ \lim_{x \to p^-} f(x) \text{ and } \lim_{x \to p^+} f(x) \text{ do not exist} \]
A classic example would be functions that have vertical asymptotes or erratic behavior within a small neighborhood of a point. Imagine trying to catch a radio station, and the signal keeps fluctuating uncontrollably, making it impossible to catch just one frequency.
One typical scenario is the behavior of the function \( f(x) = \tan(x) \) as \( x \to \frac{\pi}{2} \).

• No one-sided limits exist
• Often involves infinite discontinuities or oscillating behavior
• Graph shows no clear approach towards definite values