Problem 7 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(\lim _{x \rightarrow a+} f(x)=l=\lim _{x \rightarrow a-} g(x)\) and \(\lim _{x \rightarrow a-} f(x)=m=\) \(g(x)\), then the function \(f(x) \cdot g(x)\) (A) is not continuous at \(x=a\) (B) has a limit when \(x \rightarrow a\) and it is equal to \(l \mathrm{~m}\) (C) is continuous at \(x=a\) (D) has a limit when \(x \rightarrow a\) but it is not equal to \(l \mathrm{~m}\)

Step-by-Step Solution

Answer

The function \( f(x) \cdot g(x) \) is not continuous at \( x=a \).

Step 1: Analyze One-Sided Limits

We are given that \( \lim_{x \to a^+} f(x) = l \) and \( \lim_{x \to a^-} g(x) = l \). Both these limits converge to the same value \( l \) from opposite sides of \( a \). Similarly, \( \lim_{x \to a^-} f(x) = m \) and \( \lim_{x \to a^-} g(x) = m \) also converge to \( m \).

Step 2: Determine Limits at a

Since \( \lim_{x \to a^+} f(x) = l \) and \( \lim_{x \to a^+} g(x) = m \) converge to different values, \( f(x) \cdot g(x) \) cannot approach a single limit as \( x \) approaches \( a \).

Step 3: Test Continuity

To check the continuity, the limit of \( f(x) \cdot g(x) \) must exist and equal the function value at \( x = a \). Given the limits found essentially differ from each sided convergence, \( f(x) \cdot g(x) \) is not continuous at \( x = a \).

Step 4: Conclusion on Limit Existence

Since the behaviors of \( f(x) \) and \( g(x) \) at \( x=a \) converge to different limits from each direction, the product \( f(x) \cdot g(x) \) doesn't result in a definitive limit product, thus option (A) "is not continuous at \( x=a \)" is confirmed.

Key Concepts

One-Sided Limits

When we talk about one-sided limits, we are discussing how a function behaves as it approaches a specific point from one side only. These are crucial in understanding if a full limit exists and, subsequently, if the function is continuous. One-sided limits are essential in determining how the function approaches from either the positive or negative direction.

• Right-Hand Limit: Denoted as \( \lim_{x \to a^+} f(x) \), it considers how the function behaves as it approaches from values greater than \( a \).
• Left-Hand Limit: Denoted as \( \lim_{x \to a^-} f(x) \), it examines the approach from values less than \( a \).

Understanding both sides helps us identify potential discontinuities or if the function changes unexpectedly at that point.

Limit Existence

The existence of a limit at a point depends on one-sided limits agreeing with each other. For a function \( f(x) \) to have a limit at a point \( x = a \), the left-hand limit \( \lim_{x \to a^-} f(x) \) and the right-hand limit \( \lim_{x \to a^+} f(x) \) must both exist and be equal.
This means:

• If \( \lim_{x \to a^-} f(x) eq \lim_{x \to a^+} f(x) \), the limit does not exist at \( x=a \).
• If \( \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L \), then we say the limit exists and is \( L \).

Examining these conditions is key to testing the continuity and behavior of complex functions at specific points.

Function Behavior at Points

The behavior of a function at a specific point tells us about continuity. For a function to be continuous at a point, three conditions must be satisfied:

• \( f(a) \) is defined.
• The limit \( \lim_{x \to a} f(x) \) exists.
• The limit equals the function's value at that point, \( \lim_{x \to a} f(x) = f(a) \).

Inconsistencies in these conditions, often due to differing one-sided limits, reveal discontinuities. These breaks tell us whether a curve is neatly connected or if there are jumps and gaps, affecting the overall nature of the function.