Question 8
\[
\lim _{x \rightarrow 0}\left(\csc 2 x-\frac{1}{2 x}\right)=
\]
0
2
1
$-1$
None of these

Answer

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Answer

$\boxed{\text{None of these}}$ is the final answer because the limit does not exist

Steps

Step 1 ：Define the function as $f(x) = \csc(2x) - \frac{1}{2x}$

Step 2 ：Find the derivative of the numerator, which is $-2\cot(2x)\csc(2x) + \frac{1}{2x^2}$

Step 3 ：Find the derivative of the denominator, which is 0

Step 4 ：Calculate the limit as x approaches 0 for the ratio of the derivative of the numerator to the derivative of the denominator

Step 5 ：The limit is of the form $\frac{\infty}{0}$, which is undefined

Step 6 ：$\boxed{\text{None of these}}$ is the final answer because the limit does not exist