lim_(x rarr0)csc(x)=? Choose 1 answer: (A) -1 (B) 0 (C) 1 (D) The limit doesn't exist.

Definition of csc(x): To solve the limit of csc(x) as x approaches 0, we need to understand the definition of csc(x), which is 1/sin(x). We are looking for the limit of 1/sin(x) as x approaches 0.Behavior of sin(x) as x approaches 0: We know that sin(0) = 0. As x approaches 0, sin(x) approaches 0 as well. Therefore, we are essentially looking at the behavior of 1/0 as x approaches 0.Undefined expression 1/0: The expression 1/0 is undefined, which means that as sin(x) gets closer and closer to 0, 1/sin(x) (or csc(x)) grows without bound. This suggests that the limit does not exist because the function does not approach a finite value.Limit of csc(x) as x approaches 0: Since the limit of csc(x) as x approaches 0 does not approach a finite value and instead grows without bound, the correct answer is (D) The limit doesn't exist.

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Find derivatives of logarithmic functions

Full solution

\lim _{x \rightarrow 0} \csc (x)=?

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Choose

1

answer:

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(A)

-1

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(B)

0

\newline

(C)

1

\newline

(D) The limit doesn't exist.

Definition of $csc(x)$ : To solve the limit of $csc(x)$ as $x$ approaches $0$ , we need to understand the definition of $csc(x)$ , which is $\frac{1}{\sin(x)}$ . We are looking for the limit of $\frac{1}{\sin(x)}$ as $x$ approaches $0$ .
Behavior of sin(x) as x approaches $0$ : We know that $\sin(0) = 0$ . As $x$ approaches $0$ , $\sin(x)$ approaches $0$ as well. Therefore, we are essentially looking at the behavior of $\frac{1}{0}$ as $x$ approaches $0$ .
Undefined expression $\frac{1}{0}$ : The expression $\frac{1}{0}$ is undefined, which means that as $\sin(x)$ gets closer and closer to $0$ , $\frac{1}{\sin(x)}$ (or $\csc(x)$ ) grows without bound. This suggests that the limit does not exist because the function does not approach a finite value.
Limit of $\csc(x)$ as $x$ approaches $0$ : Since the limit of $\csc(x)$ as $x$ approaches $0$ does not approach a finite value and instead grows without bound, the correct answer is (D) The limit doesn't exist.