Which of the following functions are continuous at x=0 ? {:[f(x)=tan(x)],[g(x)=cot(x)]:} Choose 1 answer: (A) f only (B) g only (C) Both f and g (D) Neither f nor g

Question

Which of the following functions are continuous at  x=0 ?  {:[f(x)=tan(x)],[g(x)=cot(x)]:} Choose 1 answer: (A)  f only (B)  g only (C) Both  f and  g (D) Neither  f nor  g

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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) = 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) = sin(0)/cos(0) = 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 cos(x) as x approaches 0 is 1. Therefore, the limit of tan(x) as x approaches 0 is 0/1 = 0.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) = 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.

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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