Problem 207 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \lim _{x \rightarrow 3} \frac{x-3}{|x-3|} \text { . }\\{\text { Ans. } 1,-1\\} $$

Step-by-Step Solution

Answer

The limit of the function \(\frac{x - 3}{|x - 3|}\) as \(x\) approaches 3 does not exist.

Step 1 - Consider the Function for \(x < 3\)

When \(x < 3\), \(x - 3 < 0\), therefore, \(|x - 3| = -(x - 3)\). So, the function becomes \(\frac{x - 3}{-(x - 3)} = -1\) for all \(x < 3\).

Step 2 - Limit of the Function for \(x < 3\)

We calculate the left-hand limit as \(x\) approaches 3 from the left side. It's simply the constant value at all such \(x\) which implies \(\lim_{x \rightarrow 3^-} \frac{x - 3}{|x - 3|} = -1\).

Step 3 - Consider the Function for \(x > 3\)

When \(x > 3\), \(x - 3 > 0\), therefore, \(|x - 3| = x - 3\). So, the function becomes \(\frac{x - 3}{x - 3} = 1\) for all \(x > 3\).

Step 4 - Limit of the Function for \(x > 3\)

We calculate the right-hand limit as \(x\) approaches 3 from the right side. It's simply the constant value at all such \(x\) which implies \(\lim_{x \rightarrow 3^+} \frac{x - 3}{|x - 3|} = 1\).

Step 5 - Comparing Left-Hand Limit and Right-Hand Limit

The left-hand limit does not match the right-hand limit. Therefore, the limit of the function at \(x = 3\) does not exist.

Key Concepts

Continuity

Continuity is an essential concept when analyzing functions and their behavior at specific points. A function is said to be continuous at a point if its graph is unbroken at that point. In mathematical terms, a function \( f(x) \) is continuous at a point \( c \) if the following three conditions are satisfied:

• The function \( f(x) \) is defined at \( x = c \).
• The limit of \( f(x) \) as \( x \) approaches \( c \) exists.
• The limit of \( f(x) \) as \( x \) approaches \( c \) is equal to \( f(c) \).

If any of these conditions fail, the function is not continuous, which can result in a gap, jump, or vertical asymptote in its graph. In the example given, the function \( \frac{x - 3}{|x - 3|} \) shows a discontinuity at \( x = 3 \) because the left-hand limit and the right-hand limit are not equal. This inequality in limits implies a break in the graph of the function, thus indicating a point of discontinuity.

One-sided limits

Understanding one-sided limits is crucial for analyzing the behavior of functions as they get closer to a particular point from one direction only. One-sided limits include the left-hand limit and the right-hand limit:

• The left-hand limit \( \lim_{x \to c^-} f(x) \) describes the behavior of \( f(x) \) as \( x \) approaches \( c \) from the left, or lower values.
• The right-hand limit \( \lim_{x \to c^+} f(x) \) describes the behavior of \( f(x) \) as \( x \) approaches \( c \) from the right, or greater values.

To determine if the overall limit exists at a point \( c \), both the left-hand limit and right-hand limit must be equal. In the original exercise, the left-hand limit is \(-1\) and the right-hand limit is \(1\) as \( x \) approaches \( 3 \). Because these one-sided limits differ, the overall limit does not exist, emphasizing the difference in function behavior from each side of the point.

Piecewise functions

Piecewise functions are defined by different expressions over different intervals. These functions are particularly useful for representing situations where a rule or definition changes, offering flexibility in mathematical models. Each segment of the domain is handled by a specific piece of the function.
In our given problem, the function \( \frac{x - 3}{|x - 3|} \) behaves like a piecewise function around \( x = 3 \). The expression is split into two parts:

• For \( x < 3 \), the function is \( -1 \), as derived from the calculation \( \frac{x - 3}{-(x - 3)} \).
• For \( x > 3 \), the function evaluates to \( 1 \), as shown by \( \frac{x - 3}{x - 3} \).

This division into two expressions signifies different behaviors on either side of \( x = 3 \), further illustrating the concept of a piecewise function. Such functions are particularly important in real-world applications where they accurately model abrupt changes.