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lim_(x rarr(pi)/(4))cos(x)=?
Choose 1 answer:
(A)
(1)/(2)
(B) 1
(c)
(sqrt2)/(2)
(D) The limit doesn't exist.

$\lim _{x \rightarrow \frac{\pi}{4}} \cos (x)=?$ $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{1}{2}$ $\newline$ (B) $1$ $\newline$ (C) $\frac{\sqrt{2}}{2}$ $\newline$ (D) The limit doesn't exist.

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Is (x, y) a solution to the system of equations?

Full solution

Q.

\lim _{x \rightarrow \frac{\pi}{4}} \cos (x)=?

\newline

Choose

1

answer:

\newline

(A)

\frac{1}{2}

\newline

(B)

1

\newline

(C)

\frac{\sqrt{2}}{2}

\newline

(D) The limit doesn't exist.

Problem statement: We need to find the limit of the function $\cos(x)$ as $x$ approaches $\frac{\pi}{4}$ . The limit of a function at a point is the value that the function approaches as the input approaches that point. For trigonometric functions like cosine, we can directly substitute the value into the function if the function is continuous at that point and the point is within the domain of the function.
Step $1$ : Function limit definition: Since the cosine function is continuous everywhere, and $\frac{\pi}{4}$ is within its domain, we can substitute $x = \frac{\pi}{4}$ directly into the function to find the limit.
Step $2$ : Substituting $x$ into the function: Substitute $x = \frac{\pi}{4}$ into $\cos(x)$ : $\lim_{x \to \frac{\pi}{4}} \cos(x) = \cos\left(\frac{\pi}{4}\right)$
Step $3$ : Evaluating the function: We know from trigonometry that $\cos(\frac{\pi}{4})$ is equal to $\frac{\sqrt{2}}{2}$ . $\lim_{x \to \frac{\pi}{4}} \cos(x) = \frac{\sqrt{2}}{2}$
Step $4$ : Trigonometric identity: Therefore, the limit of $\cos(x)$ as $x$ approaches $\frac{\pi}{4}$ is $\frac{\sqrt{2}}{2}$ , which corresponds to choice (C).

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