Math Snap

STEP 1

Assumptions1. The function $g(x)$ is defined as $g(x)=\frac{\cos x-\sin x}{1- \sin ^{} x}$
. We are asked to find the limit of $g(x)$ as $x$ approaches $\frac{\pi}{4}$

STEP 2

First, we substitute $x=\frac{\pi}{4}$ into the function $g(x)$.
$$g\left(\frac{\pi}{4}\right)=\frac{\cos \frac{\pi}{4}-\sin \frac{\pi}{4}}{1-2 \sin ^{2} \frac{\pi}{4}}$$

STEP 3

We know that $\cos \frac{\pi}{} = \sin \frac{\pi}{} = \frac{1}{\sqrt{2}}$. Substitute these values into the equation.
$$g\left(\frac{\pi}{}\right)=\frac{\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}}{1-2 \left(\frac{1}{\sqrt{2}}\right)^{2}}$$

STEP 4

implify the numerator and the denominator separately.
$$g\left(\frac{\pi}{4}\right)=\frac{0}{1-2 \times \frac{1}{2}}$$

STEP 5

Further simplify the denominator.
$$g\left(\frac{\pi}{4}\right)=\frac{0}{1-1}$$

STEP 6

implify the fraction.
$$g\left(\frac{\pi}{4}\right)=\frac{0}{0}$$This is an indeterminate form, so we need to use L'Hopital's rule to find the limit.

STEP 7

Apply L'Hopital's rule, which states that if the limit of a function is in the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then the limit of that function is equal to the limit of the derivative of the numerator divided by the derivative of the denominator.
$$\lim{x \rightarrow \frac{\pi}{4}} g(x) = \lim{x \rightarrow \frac{\pi}{4}} \frac{g'(x)}{h'(x)}$$where $g'(x)$ is the derivative of the numerator and $h'(x)$ is the derivative of the denominator.

STEP 8

Calculate the derivative of the numerator and the denominator.
$$g'(x) = -\sin x - \cos x$$$$h'(x) = -4 \sin x \cos x$$

STEP 9

Substitute the derivatives into the limit.
$$\lim{x \rightarrow \frac{\pi}{4}} \frac{-\sin x - \cos x}{-4 \sin x \cos x}$$

STEP 10

Substitute $x=\frac{\pi}{4}$ into the limit.
$$\lim{x \rightarrow \frac{\pi}{4}} \frac{-\sin \frac{\pi}{4} - \cos \frac{\pi}{4}}{-4 \sin \frac{\pi}{4} \cos \frac{\pi}{4}}$$

STEP 11

Substitute $\sin \frac{\pi}{4} = \cos \frac{\pi}{4} = \frac{}{\sqrt{}}$ into the limit.
$$\lim{x \rightarrow \frac{\pi}{4}} \frac{-\frac{}{\sqrt{}} - \frac{}{\sqrt{}}}{-4 \times \frac{}{\sqrt{}} \times \frac{}{\sqrt{}}}$$

implify the limit.
$$\lim{x \rightarrow \frac{\pi}{4}} \frac{-\sqrt{2}}{-2} = \sqrt{2}$$So, the limit of $g(x)$ as $x$ approaches $\frac{\pi}{4}$ is $\sqrt{2}$.