g(x)=(cos(x)-sin(x))/(cos(2x)) We want to find lim_(x rarr(pi)/(4))g(x). What happens when we use direct substitution? Choose 1 answer: (A) The limit exists, and we found it! (B) The limit doesn't exist (probably an asymptote). (C) The result is indeterminate.

Direct Substitution Attempt: Let's first attempt to use direct substitution to find the limit as x approaches pi/4 for the function g(x). We substitute x with pi/4 in the function g(x): g(pi/4) = (cos(pi/4) - sin(pi/4)) / cos(2 * (pi/4))Substitute x with pi/4: Now we calculate the values of cos(pi/4), sin(pi/4), and cos(2 * (pi/4)) which are cos(pi/4) = sqrt(2)/2, sin(pi/4) = sqrt(2)/2, and cos(2 * (pi/4)) = cos(pi/2).Calculate Trigonometric Values: We know that cos(pi/2) is equal to 0. Therefore, we have: g(pi/4) = (sqrt(2)/2 - sqrt(2)/2) / 0 This simplifies to: g(pi/4) = 0 / 0Indeterminate Form: The expression 0 / 0 is an indeterminate form. This means that direct substitution does not give us the limit, and we need to use other methods to evaluate the limit.

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

g(x)=(cos(x)-sin(x))/(cos(2x))
We want to find
lim_(x rarr(pi)/(4))g(x).
What happens when we use direct substitution?
Choose 1 answer:
(A) The limit exists, and we found it!
(B) The limit doesn't exist (probably an asymptote).
(C) The result is indeterminate.

$g(x)=\frac{\cos (x)-\sin (x)}{\cos (2 x)}$ $\newline$ We want to find $\lim _{x \rightarrow \frac{\pi}{4}} g(x)$ . $\newline$ What happens when we use direct substitution? $\newline$ Choose $1$ answer: $\newline$ (A) The limit exists, and we found it! $\newline$ (B) The limit doesn't exist (probably an asymptote). $\newline$ (C) The result is indeterminate.

View step-by-step help

Full solution

g(x)=\frac{\cos (x)-\sin (x)}{\cos (2 x)}

\newline

We want to find

\lim _{x \rightarrow \frac{\pi}{4}} g(x)

\newline

What happens when we use direct substitution?

\newline

Choose

1

answer:

\newline

(A) The limit exists, and we found it!

\newline

(B) The limit doesn't exist (probably an asymptote).

\newline

Direct Substitution Attempt: Let's first attempt to use direct substitution to find the limit as $x$ approaches $\frac{\pi}{4}$ for the function $g(x)$ . We substitute $x$ with $\frac{\pi}{4}$ in the function $g(x)$ : $g(\frac{\pi}{4}) = \frac{\cos(\frac{\pi}{4}) - \sin(\frac{\pi}{4})}{\cos(2 \cdot (\frac{\pi}{4}))}$
Substitute $x$ with $\frac{\pi}{4}$ : Now we calculate the values of $\cos\left(\frac{\pi}{4}\right)$ , $\sin\left(\frac{\pi}{4}\right)$ , and $\cos\left(2 \times \left(\frac{\pi}{4}\right)\right)$ which are $\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$ , $\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$ , and $\cos\left(2 \times \left(\frac{\pi}{4}\right)\right) = \cos\left(\frac{\pi}{2}\right)$ .
Calculate Trigonometric Values: We know that $\cos(\frac{\pi}{2})$ is equal to $0$ . Therefore, we have: $\newline$ $g(\frac{\pi}{4}) = (\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}) / 0$ $\newline$ This simplifies to: $\newline$ $g(\frac{\pi}{4}) = \frac{0}{0}$
Indeterminate Form: The expression $\frac{0}{0}$ is an indeterminate form. This means that direct substitution does not give us the limit, and we need to use other methods to evaluate the limit.