Step-by-step Solution

Evaluate the limit of $x\csc\left(3x\right)$ as $x$ approaches 0

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Step-by-step explanation

Problem to solve:

$\lim_{x\to0}\left(x\cdot\csc\left(3x\right)\right)$

Learn how to solve limits by direct substitution problems step by step online.

$\lim_{x\to0}\left(\frac{\csc\left(3x\right)}{\frac{1}{x}}\right)$

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Learn how to solve limits by direct substitution problems step by step online. Evaluate the limit of xcsc(3*x) as x approaches 0. When we try to evaluate the limit directly, it results in an indeterminate form. We cannot apply L'Hôpital's rule directly, since for that we need the limit to have the form \displaystyle\frac{f(x)}{g(x)}. We can apply a little trick, which is to rewrite the function x so that the expression inside the limit is a quotient. Now, we can apply L'Hôpital's rule. Taking the derivative of cosecant. The derivative of the linear function times a constant, is equal to the constant.

Final Answer

$\frac{1}{3}$$\,\,\left(\approx 0.3333333333333333\right)$

Problem Analysis