Question

1. $limlimits_{x \to 4} \frac{\frac{1}{x} - \frac{1}{4}}{x - 4}$

Explanation:

Step1: Simplify the numerator

First, we simplify the numerator \(\frac{1}{x}-\frac{1}{4}\). To do this, we find a common denominator, which is \(4x\). So, \(\frac{1}{x}-\frac{1}{4}=\frac{4 - x}{4x}\). We can also factor out a negative sign from the numerator to get \(\frac{-(x - 4)}{4x}\).

Step2: Substitute the simplified numerator into the limit

Now, substitute this back into the original limit expression. The original limit is \(\lim_{x
ightarrow4}\frac{\frac{1}{x}-\frac{1}{4}}{x - 4}\), and after substituting the simplified numerator, we get \(\lim_{x
ightarrow4}\frac{\frac{-(x - 4)}{4x}}{x - 4}\).

Step3: Cancel out the common factor

We can cancel out the common factor of \(x - 4\) (assuming \(x
eq4\), which is valid because we are taking the limit as \(x\) approaches 4, not evaluating at \(x = 4\)). After canceling, we have \(\lim_{x
ightarrow4}\frac{-1}{4x}\).

Step4: Evaluate the limit

Now, we can evaluate the limit as \(x\) approaches 4 by substituting \(x = 4\) into the expression \(\frac{-1}{4x}\). So, \(\frac{-1}{4\times4}=-\frac{1}{16}\).

Answer:

\(-\frac{1}{16}\)