Problem 317 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \lim _{x \rightarrow 0} \frac{3 x+|x|}{7 x-5|x|} \cdot\left\\{\text { Ans. } 2, \frac{1}{6}\right\\} $$

Step-by-Step Solution

Answer

The limit of the function \(f(x) = \frac{3x + |x|}{7x - 5|x|}\) does not exist as x approaches 0, as the function exhibits different behavior when approaching 0 from the left (\(\lim_{x\to 0^-} f(x) = \frac{1}{6}\)) and from the right (\(\lim_{x\to 0^+} f(x) = 2\)).

Step 1: Define the function

First, let's define the function we're working with: \(f(x) = \frac{3x + |x|}{7x - 5|x|}\).

Step 2: Separate cases based on the sign of x

To analyze the function's behavior, we'll need to determine the value of |x| for both cases when x is positive and x is negative. If x > 0: |x| = x If x < 0: |x| = -x Now let's consider the limit as x approaches 0 from the left side, denoted as the limit with x -> 0⁻, and the limit as x approaches 0 from the right side, denoted as the limit with x -> 0⁺.

Step 3: Find the limit as x goes to 0 from the left (0⁻)

Since x is approaching 0 from the left, x is negative. Therefore, |x| = -x. \(f(x) = \frac{3x - x}{7x + 5x}\) Now let's find the limit as x approaches 0 from the left (x -> 0⁻): \(\lim_{x\to 0^-} f(x) = \lim_{x\to 0^-} \frac{2x}{12x} = \lim_{x\to 0^-} \frac{1}{6}\) The limit of f(x) as x approaches 0 from the left is \(\frac{1}{6}\).

Step 4: Find the limit as x goes to 0 from the right (0⁺)

Since x is approaching 0 from the right, x is positive. Therefore, |x| = x. \(f(x) = \frac{3x + x}{7x - 5x}\) Now let's find the limit as x approaches 0 from the right (x -> 0⁺): \(\lim_{x\to 0^+} f(x) = \lim_{x\to 0^+} \frac{4x}{2x} = \lim_{x\to 0^+} 2\) The limit of f(x) as x approaches 0 from the right is 2.

Step 5: Finalize the limit

We have found that the limit of f(x) is different when approaching 0 from the left and from the right: \(\lim_{x\to 0^-} f(x) = \frac{1}{6}\) and \(\lim_{x\to 0^+} f(x) = 2\) Due to this difference, the limit of the function as x approaches 0 does not exist. So the final answer is: The limit does not exist.

Key Concepts

Absolute Value Function

The absolute value function is a fundamental concept in calculus and many other areas of mathematics. It is denoted by \(|x|\) and defined as the distance of a number from zero on the number line.

For any real number \(x\):

• \(|x| = x\), if \(x \geq 0\)
• \(|x| = -x\), if \(x < 0\)

This function is crucial because it affects how we approach limits involving expressions with absolute values.

In the exercise given, the absolute value function \(|x|\) changes based on whether \(x\) is greater than or less than zero. This is why we break the problem into two separate cases to evaluate the limits from both the left and the right sides of the point of interest, which in this case, is zero.

Understanding how \(|x|\) behaves can significantly simplify evaluating limits and unravel situations where the function value differs when approaching from different directions.

One-Sided Limits

One-sided limits are a concept in calculus used to observe the behavior of a function as it approaches a particular point from one side — either from the left or the right.

This concept is essential when dealing with functions involving absolute values or other discontinuities.

For a function \(f(x)\), the one-sided limits are defined as:

• \(\lim_{x \to c^-} f(x)\): Limit as \(x\) approaches \(c\) from the left (negative direction).
• \(\lim_{x \to c^+} f(x)\): Limit as \(x\) approaches \(c\) from the right (positive direction).

In our exercise, the importance of one-sided limits is clear. By exploring both \(x \to 0^-\) and \(x \to 0^+\), we uncover that the limit is not the same from both directions, which ultimately informs us that the overall limit does not exist.

This highlights how one-sided limits can provide crucial information about the function's behavior around potentially problematic points.

Behavior of Functions near Points

Understanding the behavior of functions near specific points can provide valuable insights, especially when dealing with limits and continuity. The focus is primarily on how a function behaves as it approaches a particular point, which could involve changes in direction, discontinuities, or asymptotic behavior.

In our example, the function \(f(x) = \frac{3x + |x|}{7x - 5|x|}\) behaves differently as \(x\) approaches zero. This is partly due to the presence of the absolute value function in the expression. As we get close to zero:

• The function behaves significantly differently from the left and right due to the variable expression of \(|x|\).
• On one side, the function approaches \(\frac{1}{6}\), and on the other side, it tends to \(2\).

This leads to no agreement on the value of the limit at zero, demonstrating how differing behaviors on either side of a point can indicate a discontinuity or a breakdown in the expected smoothness of the function. Exploring these differences gives students deeper insight into analytical techniques used in calculus.