Questions: lim as h approaches 0 of (1/(7n + h^2)) - (1/(7h))

Solution Steps

We start with the limit expression:

\[ \lim _{h \rightarrow 0} \left(\frac{1}{7n+h^{2}}-\frac{1}{7h}\right) \]

To simplify, we find a common denominator, which is \((7n + h^2)(7h)\). This allows us to combine the fractions:

\[ \frac{7h - (7n + h^2)}{(7n + h^2)(7h)} = \frac{7h - 7n - h^2}{(7n + h^2)(7h)} \]

Next, we evaluate the limit as \( h \) approaches 0. Substituting \( h = 0 \) into the simplified expression gives:

\[ \frac{7(0) - 7n - (0)^2}{(7n + (0)^2)(7(0))} = \frac{-7n}{0} \]

This indicates that the limit approaches negative infinity, as the denominator approaches zero while the numerator remains finite.

Final Answer