Question

$\lim_{h \to 0} \frac{(5+h)^2 - 25}{h} = \square$

Explanation:

Step1: Expand the numerator

First, we expand \((5 + h)^2\) using the formula \((a + b)^2 = a^2 + 2ab + b^2\). Here, \(a = 5\) and \(b = h\), so \((5 + h)^2 = 25 + 10h + h^2\). Then the numerator \((5 + h)^2 - 25\) becomes \(25 + 10h + h^2 - 25\).
\[
(5 + h)^2 - 25=25 + 10h + h^2 - 25 = 10h + h^2
\]

Step2: Simplify the fraction

Now we have the limit \(\lim_{h
ightarrow0}\frac{10h + h^2}{h}\). We can factor out an \(h\) from the numerator: \(10h + h^2 = h(10 + h)\). Then the fraction becomes \(\frac{h(10 + h)}{h}\). Since \(h
ightarrow0\) but \(h
eq0\) (we are taking the limit as \(h\) approaches 0, not evaluating at \(h = 0\)), we can cancel out the \(h\) terms.
\[
\frac{h(10 + h)}{h}=10 + h
\]

Step3: Evaluate the limit

Now we need to find the limit as \(h\) approaches 0 of \(10 + h\). As \(h\) gets closer and closer to 0, the value of \(10 + h\) gets closer and closer to \(10+0 = 10\).
\[
\lim_{h
ightarrow0}(10 + h)=10
\]

Answer:

\(10\)