Questions: Determine the following limits. A) lim as theta approaches 0 of (1-cos(theta))/(sin(theta)) B) lim as theta approaches 0+ of (sin(theta))/(1-cos(theta)) C) lim as theta approaches 0- of (sin(theta))/(1-cos(theta)) D) lim as theta approaches 0 of (sin(theta))/(1-cos(theta))

Determine the following limits.
A) lim as theta approaches 0 of (1-cos(theta))/(sin(theta))

B) lim as theta approaches 0+ of (sin(theta))/(1-cos(theta))

C) lim as theta approaches 0- of (sin(theta))/(1-cos(theta))

D) lim as theta approaches 0 of (sin(theta))/(1-cos(theta))

Transcript text: Determine the following limits. A) $\lim _{\theta \rightarrow 0} \frac{1-\cos (\theta)}{\sin (\theta)}$ B) $\lim _{\theta \rightarrow 0^{+}} \frac{\sin (\theta)}{1-\cos (\theta)}$ C) $\lim _{\theta \rightarrow 0^{-}} \frac{\sin (\theta)}{1-\cos (\theta)}$ D) $\lim _{\theta \rightarrow 0} \frac{\sin (\theta)}{1-\cos (\theta)}$

Solution

Solution Steps

To determine the limits, we can use L'Hôpital's Rule, which is applicable when the limit results in an indeterminate form like 0/0 or ∞/∞. For each limit, we will differentiate the numerator and the denominator and then evaluate the limit.

A) $\lim _{\theta \rightarrow 0} \frac{1-\cos (\theta)}{\sin (\theta)}$

Apply L'Hôpital's Rule since the limit is in the form 0/0.
Differentiate the numerator and the denominator.
Evaluate the limit of the resulting expression.

B) $\lim _{\theta \rightarrow 0^{+}} \frac{\sin (\theta)}{1-\cos (\theta)}$

Apply L'Hôpital's Rule since the limit is in the form 0/0.
Differentiate the numerator and the denominator.
Evaluate the limit of the resulting expression.

C) $\lim _{\theta \rightarrow 0^{-}} \frac{\sin (\theta)}{1-\cos (\theta)}$

Apply L'Hôpital's Rule since the limit is in the form 0/0.
Differentiate the numerator and the denominator.
Evaluate the limit of the resulting expression.

Step 1: Determine the limit for part A

To find $\lim _{\theta \rightarrow 0} \frac{1-\cos (\theta)}{\sin (\theta)}$, we apply L'Hôpital's Rule because the limit is in the form $\frac{0}{0}$.

Differentiate the numerator: $\frac{d}{d\theta}(1 - \cos(\theta)) = \sin(\theta)$
Differentiate the denominator: $\frac{d}{d\theta}(\sin(\theta)) = \cos(\theta)$

Thus, the limit becomes: \[ \lim _{\theta \rightarrow 0} \frac{\sin(\theta)}{\cos(\theta)} = \frac{\sin(0)}{\cos(0)} = \frac{0}{1} = 0 \]

Step 2: Determine the limit for part B

To find $\lim _{\theta \rightarrow 0^{+}} \frac{\sin (\theta)}{1-\cos (\theta)}$, we apply L'Hôpital's Rule because the limit is in the form $\frac{0}{0}$.

Differentiate the numerator: $\frac{d}{d\theta}(\sin(\theta)) = \cos(\theta)$
Differentiate the denominator: $\frac{d}{d\theta}(1 - \cos(\theta)) = \sin(\theta)$

Thus, the limit becomes: \[ \lim _{\theta \rightarrow 0^{+}} \frac{\cos(\theta)}{\sin(\theta)} = \frac{\cos(0)}{\sin(0)} = \frac{1}{0} = +\infty \]

Step 3: Determine the limit for part C

To find $\lim _{\theta \rightarrow 0^{-}} \frac{\sin (\theta)}{1-\cos (\theta)}$, we apply L'Hôpital's Rule because the limit is in the form $\frac{0}{0}$.

Differentiate the numerator: $\frac{d}{d\theta}(\sin(\theta)) = \cos(\theta)$
Differentiate the denominator: $\frac{d}{d\theta}(1 - \cos(\theta)) = \sin(\theta)$

Thus, the limit becomes: \[ \lim _{\theta \rightarrow 0^{-}} \frac{\cos(\theta)}{\sin(\theta)} = \frac{\cos(0)}{\sin(0)} = \frac{1}{0} = -\infty \]