Question

If \( f(x) = x - \frac{1}{x} \), the value of 

• A. 0
• B. \( \frac{1}{2} \)
• C. 1
• D. 2

Hint:

For limits involving functions with \( \Delta x \), simplify the expression carefully and compute the result as \( \Delta x \) approaches 0.

Answer 1

The Correct Option is D

To solve for the limit, we first compute \( f(1 + \Delta x) \) and \( f(1) \):
\( f(1 + \Delta x) = (1 + \Delta x) - \frac{1}{1 + \Delta x} \)
\( f(1) = 1 - \frac{1}{1} = 0 \)
Now, we substitute these into the limit expression: \[ \lim_{\Delta x \to 0} \frac{(1 + \Delta x) - \frac{1}{1 + \Delta x} - 0}{\Delta x} = \lim_{\Delta x \to 0} \frac{(1 + \Delta x) - \frac{1}{1 + \Delta x}}{\Delta x} \] Simplifying the numerator: \[ (1 + \Delta x) - \frac{1}{1 + \Delta x} = \frac{(1 + \Delta x)^2 - 1}{1 + \Delta x} = \frac{\Delta x(2 + \Delta x)}{1 + \Delta x} \] Now, the expression becomes: \[ \lim_{\Delta x \to 0} \frac{\Delta x(2 + \Delta x)}{\Delta x(1 + \Delta x)} = \lim_{\Delta x \to 0} \frac{2 + \Delta x}{1 + \Delta x} \] As \( \Delta x \to 0 \), the limit evaluates to: \[ \frac{2}{1} = 2 \] Thus, the value of the limit is \( 2 \).