To evaluate the limit [tex]\(\lim_{h \to a} \frac{\frac{1}{h} - \frac{1}{a}}{h - a}\)[/tex], let's go through a step-by-step solution.
1. Given Expression:
[tex]\[
\lim_{h \to a} \frac{\frac{1}{h} - \frac{1}{a}}{h - a}
\][/tex]
2. Combine the Fractions in the Numerator:
To simplify, we need to combine the fractions in the numerator [tex]\(\frac{1}{h} - \frac{1}{a}\)[/tex]:
[tex]\[
\frac{1}{h} - \frac{1}{a} = \frac{a - h}{ah}
\][/tex]
Thus, the expression becomes:
[tex]\[
\lim_{h \to a} \frac{\frac{a - h}{ah}}{h - a}
\][/tex]
3. Simplify the Fraction:
The numerator of our limit is:
[tex]\[
\frac{a - h}{ah}
\][/tex]
and we divide it by [tex]\(h - a\)[/tex]. Notice that:
[tex]\[
\frac{\frac{a - h}{ah}}{h - a} = \frac{a - h}{ah(h - a)}
\][/tex]
Since [tex]\(a - h = -(h - a)\)[/tex], this can be rewritten as:
[tex]\[
\frac{-(h - a)}{ah(h - a)} = \frac{-1}{ah}
\][/tex]
4. Calculate the Limit:
Now, we evaluate the limit as [tex]\(h\)[/tex] approaches [tex]\(a\)[/tex]:
[tex]\[
\lim_{h \to a} \frac{-1}{ah} = \frac{-1}{a \cdot a} = \frac{-1}{a^2}
\][/tex]
Therefore, the final result is:
[tex]\[
\lim_{h \to a} \frac{\frac{1}{h} - \frac{1}{a}}{h - a} = \frac{-1}{a^2}
\][/tex]
