lim_(x rarr pi)tan(x)=? Choose 1 answer: (A) -1 (B) 0 (C) 1 (D) The limit doesn't exist.

Step 1: Evaluate tangent function near x = pi: To find the limit of tan(x) as x approaches pi, we need to evaluate the behavior of the tangent function near x = pi.Step 2: Definition of tangent function: The tangent function, tan(x), is the ratio of the sine function to the cosine function, so tan(x) = sin(x)/cos(x).Step 3: Behavior of sin(x) as x approaches pi: As x approaches pi, sin(x) approaches 0 because sin(pi) = 0.Step 4: Behavior of cos(x) as x approaches pi: As x approaches pi, cos(x) approaches -1 because cos(pi) = -1.Step 5: Limit of tan(x) as x approaches pi: Therefore, the limit of tan(x) as x approaches pi is the limit of sin(x)/cos(x) as x approaches pi, which is 0/(-1).Step 6: Final result: The limit of 0 divided by any non-zero number is 0. Since we are dividing by -1, which is non-zero, the limit is 0.

Home

Math Problems

Calculus

Find derivatives of logarithmic functions

Full solution

\lim _{x \rightarrow \pi} \tan (x)=?

\newline

Choose

1

answer:

\newline

(A)

-1

\newline

(B)

0

\newline

(C)

1

\newline

(D) The limit doesn't exist.

Step $1$ : Evaluate tangent function near $x = \pi$ : To find the limit of $\tan(x)$ as $x$ approaches $\pi$ , we need to evaluate the behavior of the tangent function near $x = \pi$ .
Step $2$ : Definition of tangent function: The tangent function, $\tan(x)$ , is the ratio of the sine function to the cosine function, so $\tan(x) = \frac{\sin(x)}{\cos(x)}$ .
Step $3$ : Behavior of $\sin(x)$ as $x$ approaches $\pi$ : As $x$ approaches $\pi$ , $\sin(x)$ approaches $0$ because $\sin(\pi) = 0$ .
Step $4$ : Behavior of $\cos(x)$ as $x$ approaches $\pi$ : As $x$ approaches $\pi$ , $\cos(x)$ approaches $-1$ because $\cos(\pi) = -1$ .
Step $5$ : Limit of $\tan(x)$ as $x$ approaches $\pi$ : Therefore, the limit of $\tan(x)$ as $x$ approaches $\pi$ is the limit of $\sin(x)/\cos(x)$ as $x$ approaches $\pi$ , which is $0/(-1)$ .
Step $6$ : Final result: The limit of $0$ divided by any non-zero number is $0$ . Since we are dividing by $-1$ , which is non-zero, the limit is $0$ .