To find the limit [tex]\(\lim _{h \rightarrow 0} \frac{(2+h)^2 - 4}{h}\)[/tex], let's go through the solution step-by-step.
1. Substitute and Expand the Expression:
First, let's expand the expression inside the limit:
[tex]\[
(2 + h)^2
\][/tex]
Expanding this, we get:
[tex]\[
(2 + h)^2 = 2^2 + 2 \cdot 2 \cdot h + h^2 = 4 + 4h + h^2
\][/tex]
2. Simplify the Expression:
Now plug the expanded form back into the original limit expression:
[tex]\[
\frac{(2 + h)^2 - 4}{h} = \frac{4 + 4h + h^2 - 4}{h}
\][/tex]
Simplify the numerator:
[tex]\[
4 + 4h + h^2 - 4 = 4h + h^2
\][/tex]
The expression is now:
[tex]\[
\frac{4h + h^2}{h}
\][/tex]
3. Factor and Simplify Further:
We can factor out [tex]\(h\)[/tex] from the numerator:
[tex]\[
\frac{h(4 + h)}{h}
\][/tex]
Now cancel [tex]\(h\)[/tex] from the numerator and the denominator:
[tex]\[
4 + h
\][/tex]
4. Take the Limit:
Finally, take the limit as [tex]\(h\)[/tex] approaches 0:
[tex]\[
\lim _{h \rightarrow 0} (4 + h)
\][/tex]
Substituting [tex]\(h = 0\)[/tex] into the expression, we get:
[tex]\[
4 + 0 = 4
\][/tex]
Therefore, the limit is:
[tex]\[
\lim _{h \rightarrow 0} \frac{(2+h)^2 - 4}{h} = 4
\][/tex]
