Questions: lim as h approaches 0 of ((9+h)^2-81)/h

Solution Steps

To solve the limit problem, we need to simplify the expression inside the limit. First, expand the numerator and then simplify the fraction. Finally, take the limit as \( h \) approaches 0.

First, we expand the expression \((9 + h)^2\): \[ (9 + h)^2 = 81 + 18h + h^2 \]

Next, we substitute the expanded form back into the original limit expression: \[ \frac{(9 + h)^2 - 81}{h} = \frac{81 + 18h + h^2 - 81}{h} = \frac{18h + h^2}{h} \]

We can factor \( h \) out of the numerator: \[ \frac{18h + h^2}{h} = \frac{h(18 + h)}{h} = 18 + h \]

Finally, we take the limit of the simplified expression as \( h \) approaches 0: \[ \lim_{h \to 0} (18 + h) = 18 \]

Final Answer