Question

evaluate the limit
lim_{a
ightarrow4}\frac{\frac{1}{a}-\frac{1}{4}}{a - 4}
\frac{1}{8} ×
enter a mathematical expression more...
question help: video

Explanation:

Step1: Combine fractions in numerator

First, combine $\frac{1}{a}-\frac{1}{4}=\frac{4 - a}{4a}$. So the limit becomes $\lim_{a
ightarrow4}\frac{\frac{4 - a}{4a}}{a - 4}$.

Step2: Simplify the complex - fraction

$\lim_{a
ightarrow4}\frac{\frac{4 - a}{4a}}{a - 4}=\lim_{a
ightarrow4}\frac{4 - a}{4a(a - 4)}$. Factor out - 1 from the numerator: $\lim_{a
ightarrow4}\frac{-(a - 4)}{4a(a - 4)}$.

Step3: Cancel out common factors

Cancel out the common factor $(a - 4)$ (since $a
eq4$ when taking the limit), we get $\lim_{a
ightarrow4}-\frac{1}{4a}$.

Step4: Evaluate the limit

Substitute $a = 4$ into $-\frac{1}{4a}$, we have $-\frac{1}{4\times4}=-\frac{1}{16}$.

Answer:

$-\frac{1}{16}$