Question

1. use lhopital to find the following limits.

a. \\( \lim _ { h \
ightarrow 0 } \frac { ( 1 + h ) ^ { - 2 } - 1 } { h } \\)
b. \\( \lim _ { \theta \
ightarrow \pi } \frac { \sin \theta } { \theta - \pi } \\)
f. \\( \lim _ { x \
ightarrow 0 } \frac { x - \sin x } { x ^ { 3 } + x ^ { 2 } } \\)
g. \\( \lim _ { x \
ightarrow 0 } ( 1 + 2 x ) ^ { 1 / x } \\)

Explanation:

Step1: Check the form of the limit

For \(\lim_{h
ightarrow0}\frac{(1 + h)^{-2}-1}{h}\), when \(h = 0\), the form is \(\frac{0}{0}\).

Step2: Apply L'Hopital's Rule

Differentiate the numerator and denominator.
The derivative of \((1 + h)^{-2}-1\) is \(- 2(1 + h)^{-3}\), and the derivative of \(h\) is \(1\).
So \(\lim_{h
ightarrow0}\frac{(1 + h)^{-2}-1}{h}=\lim_{h
ightarrow0}-2(1 + h)^{-3}\)

Step3: Evaluate the limit

Substitute \(h = 0\) into \(-2(1 + h)^{-3}\), we get \(-2(1+0)^{-3}=-2\)

Answer:

\(-2\)