Question

Find the limit (if it exists). If it does not exist, explain why. $$ \lim _{\Delta x \rightarrow 0^{-}} \frac{\frac{1}{x+\Delta x}-\frac{1}{x}}{\Delta x} $$

Short answer

The limit as \(\Delta x\) approaches 0 is \(-\frac{1}{x^2}\).

Step-by-step solution

Simplify the Expression

First, we simplify the expression in the numerator. We can combine the fractions \(\frac{1}{x+\Delta x}\) and \(\frac{1}{x}\) under a common denominator, which will be \(x(x+\Delta x)\). The simplified form will look like this: \(-\frac{\Delta x}{x(x+\Delta x)}\). Now, we have: \(\lim _{\Delta x \rightarrow 0^{-}} -\frac{\Delta x}{x(x+\Delta x) \Delta x}\).

Further Simplify the Expression by Cancelling Out

We can cancel out \(\Delta x\) in the numerator and denominator. After this, we have: \(\lim _{\Delta x \rightarrow 0^{-}} -\frac{1}{x(x+\Delta x)}\).

Evaluate the Limit

Now, we substitute \(\Delta x\) with 0 directly in the equation to find the limit. This leads to:\(-\frac{1}{x(x+0)}\) or \(-\frac{1}{x^2}\).