Problem 39 - A Complete Resource Book in Mathematics for JEE Main

Question

\(\lim _{x \rightarrow e} \frac{\ln x-1}{|x-e|}\) is equal to (A) \(\frac{1}{e}\) (B) \(-\frac{1}{e}\) (C) \(e\) (D) Does not exist

Step-by-Step Solution

Answer

The limit does not exist: option (D).

Step 1: Analyze the Expression

We need to evaluate the limit of \(\lim_{x \to e} \frac{\ln x - 1}{|x - e|}\). The numerator \(\ln x - 1\) implies we are dealing with a logarithmic function, while the denominator \(|x - e|\) suggests we need to consider the limit from both sides of \(x = e\).

Step 2: Simplifying the Numerator

For \(x\) approaching \(e\), the expression \(\ln x - 1\) can be written as \(\ln x - \ln e = \ln\left(\frac{x}{e}\right)\). This transformation can help in simplifying the expression by substituting \(u = x/e\) so that \(\ln u\) when \(x \to e\).

Step 3: Approach the Limit from the Right

We first consider the case where \(x \to e^+\). As \(x\) approaches \(e\) from the right, \(|x - e| = x - e\). Therefore, the limit becomes \(\lim_{x \to e^+} \frac{\ln \left(\frac{x}{e}\right)}{x-e}\).

Step 4: Derivative Using L'Hôpital's Rule (Right)

Since this limit yields the indeterminate form \(\frac{0}{0}\), apply L'Hôpital's Rule: differentiate the numerator and the denominator. The derivative of \(\ln\left(\frac{x}{e}\right)\) is \(\frac{1}{x}\), and the derivative of \(x-e\) is 1. Thus, the limit becomes \(\lim_{x \to e^+} \frac{1/x}{1} = 1/e\).

Step 5: Approach the Limit from the Left

Next, consider \(x \to e^-\). Here, \(|x - e| = e - x\). Thus, the limit transforms to \(\lim_{x \to e^-} \frac{\ln \left(\frac{x}{e}\right)}{e-x}\).

Step 6: Derivative Using L'Hôpital's Rule (Left)

This limit also results in the indeterminate form \(\frac{0}{0}\), so apply L'Hôpital's Rule again by differentiating. The derivative of the numerator remains \(\frac{1}{x}\), and the derivative of \(e-x\) is \(-1\). Therefore, the limit becomes \(\lim_{x \to e^-} \frac{1/x}{-1} = -1/e\).

Step 7: Evaluate the Two-Sided Limit

For \(\lim_{x \to e} \frac{\ln x - 1}{|x - e|}\) to exist, limits from the left and right must be equal. Since \(1/e\) from the right does not equal \(-1/e\) from the left, the two-sided limit does not exist.

Key Concepts

L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool in calculus used to evaluate limits involving indeterminate forms such as \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). These forms occur frequently when both the numerator and denominator of a fraction approach zero or infinity as the variable approaches a particular value. L'Hôpital's Rule helps simplify these situations by allowing us to differentiate the numerator and denominator separately.

Here's how to use L'Hôpital's Rule effectively:

• Ensure the expression you're dealing with is indeed in an indeterminate form like \(\frac{0}{0}\).
• Check that both the numerator and denominator are differentiable in the vicinity of the point of interest.
• Apply the rule by taking the derivative of the numerator and the derivative of the denominator separately.
• Compute the limit of the resulting expression.

In the given exercise, we applied L'Hôpital's Rule to the limits from both the left and right of \(x = e\), since both sides yielded \(\frac{0}{0}\) forms. Differentiating appropriately allowed us to determine the behavior of the function as \(x\) approached \(e\) from both sides.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. In simpler terms, if you know \(b^y = x\), then \(\log_b(x) = y\). The common logarithmic function base is the natural logarithm (base \(e\)), written as \(\ln x\). This is the logarithm used in the exercise.

Key properties of logarithmic functions include:

• The logarithm of 1 is always 0: \(\ln 1 = 0\).
• The logarithm of a product can be expressed as the sum of logarithms: \(\ln(xy) = \ln x + \ln y\).
• The logarithm of a quotient can be expressed as the difference of logarithms: \(\ln\left(\frac{x}{y}\right) = \ln x - \ln y\).

In the limit problem, the expression \(\ln x - 1\) was transformed using logarithmic properties into a simpler form: \(\ln\left(\frac{x}{e}\right)\). This made it easier to apply L'Hôpital's Rule by clearly defining the indeterminate form.

Two-Sided Limits

Two-sided limits evaluate the behavior of a function as the variable approaches a particular point from both directions, left and right. For a limit to exist at a point, the limits from both sides must be equal.

Here's how you can think about two-sided limits:

• If \(\lim_{x \to c^+} f(x) = \lim_{x \to c^-} f(x)\), then \(\lim_{x \to c} f(x)\) exists and equals that common value.
• If the left-hand limit (approaching from the left) and the right-hand limit (approaching from the right) are not equal, the two-sided limit does not exist.

In the exercise, we evaluated both \(\lim_{x \to e^+} \) and \(\lim_{x \to e^-} \) of the given expression. Since these limits resulted in different values, specifically \(\frac{1}{e}\) and \(-\frac{1}{e}\), respectively, the two-sided limit at \(x = e\) does not exist. This highlights the importance of considering both sides when dealing with limits involving absolute values or direction-dependent expressions.