To find the limit \(\lim _{h \rightarrow 0} h^2\left(1+\frac{1}{h^2}\right)\), follow these steps:
1. Express the function \(f(h)\):
[tex]\[
f(h) = h^2 \left(1 + \frac{1}{h^2}\right)
\][/tex]
2. Simplify the expression inside the limit:
[tex]\[
f(h) = h^2 \left(1 + \frac{1}{h^2}\right)
\][/tex]
Distribute \(h^2\) across the terms inside the parentheses:
[tex]\[
f(h) = h^2 \cdot 1 + h^2 \cdot \frac{1}{h^2}
\][/tex]
This simplifies to:
[tex]\[
f(h) = h^2 + 1
\][/tex]
3. Evaluate the limit as \(h\) approaches 0:
Now, we need to find the limit of the simplified expression:
[tex]\[
\lim_{h \to 0} (h^2 + 1)
\][/tex]
4. Substitute \(h = 0\) in the simplified function:
As \(h\) approaches 0:
[tex]\[
h^2 \to 0^2 = 0
\][/tex]
Therefore:
[tex]\[
h^2 + 1 \to 0 + 1 = 1
\][/tex]
So, the limit is:
[tex]\[
\lim_{h \to 0} h^2 \left(1 + \frac{1}{h^2}\right) = 1
\][/tex]
