Question 11 Evaluate limθ→ 0 f(θ), where f(θ) =...

Transcript

00:01 We are looking for the limit as theta approaches 0 of cosine squared theta minus 1 over theta.

00:15 The first thing you want to do is to apply loputhythal's rule.

00:19 If our limit as x approaches a, a in this case is 0, of f of x over g of x is equal to 0 over 0.

00:35 Or plus minus infinity over plus minus infinity, then the limit as x approaches a of f of x over g of x will be equal to its derivative.

00:55 We're also notated as the limit as x approaches a of f prime of x over g prime of x.

01:07 The first thing we want to do is to test for the 0 over 0 condition of lopithel's rule.

01:16 We are going to take the numerator and denominator and make them equal to 0.

01:22 So the limit as theta approaches 0 of cosine squared theta minus 1 is equal to 0.

01:35 Replace theta with 0, we get cosine squared 0 minus 1 is equal to 0.

01:45 We are going to use the trivial identity of cosine of zero being equal to 1.

01:54 Applying this rule, we will get 1 minus 1 being equal to 0.

02:07 The math checks out.

02:09 Now we are going to make the denominator equal to 0.

02:13 So the limit as theta approaches 0 of theta is equal to 0.

02:22 Replace theta with 0, we get 0 is equal to 0.

02:27 This also checks out.

02:30 So now that we know that our limit can be fulfilled by lopi doll's rule, now we can take the derivative of the limit.

02:42 So it will look like the limit as theta approaches 0 of cosine squared theta minus 1, all of that prime over theta prime...

Question 11 Evaluate $\lim_{\theta \to 0} f(\theta)$, where $f(\theta) = \frac{\sin\theta \cdot \cos3\theta}{\theta \cdot (1 + \cos2\theta)}$. 10 Points