Calculus Examples

$lim_{x \to 0} \frac{6}{4 x (\csc (6 x))}$

Step 1

Cancel the common factor of $6$ and $4$ .

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Step 1.1

Factor $2$ out of $6$ .

$lim_{x \to 0} \frac{2 (3)}{4 x (\csc (6 x))}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $2$ out of $4 x (\csc (6 x))$ .

$lim_{x \to 0} \frac{2 (3)}{2 (2 x \csc (6 x))}$

Step 1.2.2

Cancel the common factor.

$lim_{x \to 0} \frac{2 \cdot 3}{2 (2 x \csc (6 x))}$

Step 1.2.3

Rewrite the expression.

$lim_{x \to 0} \frac{3}{2 x \csc (6 x)}$

Step 2

Move the term $\frac{3}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{3}{2} lim_{x \to 0} \frac{1}{x \csc (6 x)}$

Step 3

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $0$ .

$\frac{3}{2} \cdot \frac{lim_{x \to 0} 1}{lim_{x \to 0} x \csc (6 x)}$

Step 4

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\frac{3}{2} \cdot \frac{1}{lim_{x \to 0} x \csc (6 x)}$

Step 5

Consider the left sided limit.

$lim_{x \to 0^{-}} x \csc (6 x)$

Step 6

Make a table to show the behavior of the function $x \csc (6 x)$ as $x$ approaches $0$ from the left.

$\begin{array}{c} x & x \csc (6 x) \\ - 0.1 & 0.17710321 \\ - 0.01 & 0.1667667 \\ - 0.001 & 0.16666766 \end{array}$

Step 7

As the $x$ values approach $0$ , the function values approach $0.167$ . Thus, the limit of $x \csc (6 x)$ as $x$ approaches $0$ from the left is $0.167$ .

$0.167$

Step 8

Consider the right sided limit.

$lim_{x \to 0^{+}} x \csc (6 x)$

Step 9

Make a table to show the behavior of the function $x \csc (6 x)$ as $x$ approaches $0$ from the right.

$\begin{array}{c} x & x \csc (6 x) \\ 0.1 & 0.17710321 \\ 0.01 & 0.1667667 \\ 0.001 & 0.16666766 \end{array}$

Step 10

As the $x$ values approach $0$ , the function values approach $0.167$ . Thus, the limit of $x \csc (6 x)$ as $x$ approaches $0$ from the right is $0.167$ .

$\frac{3}{2} \cdot \frac{1}{0.167}$

Step 11

Simplify the answer.

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Step 11.1

Divide $1$ by $0.167$ .

$\frac{3}{2} \cdot 5.98802395$

Step 11.2

Multiply $\frac{3}{2} \cdot 5.98802395$ .

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Step 11.2.1

Combine $\frac{3}{2}$ and $5.98802395$ .

$\frac{3 \cdot 5.98802395}{2}$

Step 11.2.2

Multiply $3$ by $5.98802395$ .

$\frac{17.96407185}{2}$

Step 11.3

Divide $17.96407185$ by $2$ .

$8.98203592$