Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Consider the left sided limit.

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

Step 4

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

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

Step 5

As the $x$ values approach $0$ , the function values approach $1$ . Thus, the limit of $x \csc (x)$ as $x$ approaches $0$ from the left is $1$ .

$1$

Step 6

Consider the right sided limit.

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

Step 7

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

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

Step 8

As the $x$ values approach $0$ , the function values approach $1$ . Thus, the limit of $x \csc (x)$ as $x$ approaches $0$ from the right is $1$ .

$\frac{1 + 1}{x \csc (x)}$

Step 9

Simplify the answer.

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Step 9.1

Add $1$ and $1$ .

$\frac{2}{x \csc (x)}$

Step 9.2

Separate fractions.

$\frac{2}{x} \cdot \frac{1}{\csc (x)}$

Step 9.3

Rewrite $\csc (x)$ in terms of sines and cosines.

$\frac{2}{x} \cdot \frac{1}{\frac{1}{\sin (x)}}$

Step 9.4

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (x)}$ .

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

Step 9.5

Multiply $\sin (x)$ by $1$ .

$\frac{2}{x} \sin (x)$

Step 9.6

Combine $\frac{2}{x}$ and $\sin (x)$ .

$\frac{2 \sin (x)}{x}$