Calculus Examples

lim x \to 0 \frac{2}{x csc (x)}

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

Step 1

Move the term

2

$2$ outside of the limit because it is constant with respect to

x

$x$ .

2 lim x \to 0 \frac{1}{x csc (x)}

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

Step 2

Split the limit using the Limits Quotient Rule on the limit as

x

$x$ approaches

0

$0$ .

2 \frac{lim x \to 0 1}{lim x \to 0 x csc (x)}

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

Step 3

Evaluate the limit of

1

$1$ which is constant as

x

$x$ approaches

0

$0$ .

2 \frac{1}{lim x \to 0 x csc (x)}

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

Step 4

Consider the left sided limit.

lim x \to 0^{-} x csc (x)

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

Step 5

Make a table to show the behavior of the function

x csc (x)

$x \csc (x)$ as

x

$x$ approaches

0

$0$ from the left.

\begin{matrix} x & x csc (x) - 0.1 & 1.00166861 - 0.01 & 1.00001666 - 0.001 & 1.00000016 \end{matrix}

$\begin{array}{c} x & x \csc (x) \\ - 0.1 & 1.00166861 \\ - 0.01 & 1.00001666 \\ - 0.001 & 1.00000016 \end{array}$

Step 6

As the

x

$x$ values approach

0

$0$ , the function values approach

1

$1$ . Thus, the limit of

x csc (x)

$x \csc (x)$ as

x

$x$ approaches

0

$0$ from the left is

1

$1$ .

1

$1$

Step 7

Consider the right sided limit.

lim x \to 0^{+} x csc (x)

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

Step 8

Make a table to show the behavior of the function

x csc (x)

$x \csc (x)$ as

x

$x$ approaches

0

$0$ from the right.

\begin{matrix} x & x csc (x) 0.1 & 1.00166861 0.01 & 1.00001666 0.001 & 1.00000016 \end{matrix}

$\begin{array}{c} x & x \csc (x) \\ 0.1 & 1.00166861 \\ 0.01 & 1.00001666 \\ 0.001 & 1.00000016 \end{array}$

Step 9

As the

x

$x$ values approach

0

$0$ , the function values approach

1

$1$ . Thus, the limit of

x csc (x)

$x \csc (x)$ as

x

$x$ approaches

0

$0$ from the right is

1

$1$ .

2 (\frac{1}{1})

$2 (\frac{1}{1})$

Step 10

Simplify the answer.

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Step 10.1

Cancel the common factor of

1

$1$ .

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Step 10.1.1

Cancel the common factor.

$2 (\frac{1}{1})$

Step 10.1.2

Rewrite the expression.

$2 \cdot 1$

Step 10.2

Multiply

$2$ by

$1$ .

$2$