Which of the following functions are continuous at $x=0$ ? $\newline$ $\begin{array}{l} f(x)=\tan (x) \\ g(x)=\cot (x) \end{array}$ $\newline$ Choose $1$ answer: $\newline$ (A) $f$ only $\newline$ (B) $g$ only $\newline$ (C) Both $f$ and $g$ $\newline$ (D) Neither $f$ nor $g$

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Algebra 2

Domain and range of absolute value functions: equations

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Q. Which of the following functions are continuous at

x=0

\newline

\begin{array}{l} f(x)=\tan (x) \\ g(x)=\cot (x) \end{array}

\newline

Choose

1

answer:

\newline

(A)

f

only

\newline

(B)

g

only

\newline

f

and

g

\newline

(D) Neither

f

nor

g

Function Continuity Check: To determine if a function is continuous at a point, we need to check if the function is defined at that point, if the limit exists at that point, and if the limit equals the function's value at that point.
Consideration of $f(x) = \tan(x)$ : Let's first consider $f(x) = \tan(x)$ . The tangent function is the ratio of the sine function to the cosine function, $\tan(x) = \frac{\sin(x)}{\cos(x)}$ . At $x = 0$ , both sine and cosine are defined, with $\sin(0) = 0$ and $\cos(0) = 1$ .
Definition of $f(x)$ at $x = 0$ : Since $\tan(0) = \frac{\sin(0)}{\cos(0)} = \frac{0}{1} = 0$ , the function $f(x) = \tan(x)$ is defined at $x = 0$ . Now we need to check the limit of $\tan(x)$ as $x$ approaches $0$ .
Limit of $\tan(x)$ as $x$ approaches $0$ : The limit of $\tan(x)$ as $x$ approaches $0$ is $0$ , because the limit of $\sin(x)$ as $x$ approaches $0$ is $0$ and the limit of $x$ $1$ as $x$ approaches $0$ is $x$ $4$ . Therefore, the limit of $\tan(x)$ as $x$ approaches $0$ is $x$ $8$ .
Continuity of $f(x)$ at $x = 0$ : Since the limit of $f(x)$ as $x$ approaches $0$ is equal to $f(0)$ , the function $f(x) = \tan(x)$ is continuous at $x = 0$ .
Consideration of $g(x) = \cot(x)$ : Now let's consider $g(x) = \cot(x)$ . The cotangent function is the reciprocal of the tangent function, $\cot(x) = \frac{\cos(x)}{\sin(x)}$ . At $x = 0$ , the sine function is $0$ , which would make the denominator of $\cot(x)$ equal to $0$ .
Undefined at $x = 0$ : Since division by zero is undefined, the function $g(x) = \cot(x)$ is not defined at $x = 0$ . Therefore, $g(x)$ cannot be continuous at $x = 0$ because it is not defined at that point.
Continuity analysis: Based on the analysis, $f(x) = \tan(x)$ is continuous at $x = 0$ , while $g(x) = \cot(x)$ is not continuous at $x = 0$ . Therefore, the correct answer is (A) $f$ only.