Calculus Examples

$lim_{x \to 0} \frac{x \csc (2 x)}{\cos (5 x)}$

Step 1

Evaluate the limit.

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Step 1.1

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

$\frac{lim_{x \to 0} x \csc (2 x)}{lim_{x \to 0} \cos (5 x)}$

Step 1.2

Move the limit inside the trig function because cosine is continuous.

$\frac{lim_{x \to 0} x \csc (2 x)}{\cos (lim_{x \to 0} 5 x)}$

Step 1.3

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{lim_{x \to 0} x \csc (2 x)}{\cos (5 lim_{x \to 0} x)}$

Step 2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{lim_{x \to 0} x \csc (2 x)}{\cos (5 \cdot 0)}$

Step 3

Consider the left sided limit.

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

Step 4

Make a table to show the behavior of the function $x \csc (2 x)$ as $x$ approaches $0$ from the left.

$\begin{array}{c} x & x \csc (2 x) \\ - 0.1 & 0.50334895 \\ - 0.01 & 0.50003333 \\ - 0.001 & 0.50000033 \end{array}$

Step 5

As the $x$ values approach $0$ , the function values approach $0.5$ . Thus, the limit of $x \csc (2 x)$ as $x$ approaches $0$ from the left is $0.5$ .

$0.5$

Step 6

Consider the right sided limit.

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

Step 7

Make a table to show the behavior of the function $x \csc (2 x)$ as $x$ approaches $0$ from the right.

$\begin{array}{c} x & x \csc (2 x) \\ 0.1 & 0.50334895 \\ 0.01 & 0.50003333 \\ 0.001 & 0.50000033 \end{array}$

Step 8

As the $x$ values approach $0$ , the function values approach $0.5$ . Thus, the limit of $x \csc (2 x)$ as $x$ approaches $0$ from the right is $0.5$ .

$\frac{0.5}{\cos (5 \cdot 0)}$

Step 9

Simplify the answer.

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Step 9.1

Separate fractions.

$\frac{0.5}{1} \cdot \frac{1}{\cos (5 \cdot 0)}$

Step 9.2

Convert from $\frac{1}{\cos (5 \cdot 0)}$ to $\sec (5 \cdot 0)$ .

$\frac{0.5}{1} \sec (5 \cdot 0)$

Step 9.3

Multiply $5$ by $0$ .

$\frac{0.5}{1} \sec (0)$

Step 9.4

Cancel the common factor of $0.5$ .

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Step 9.4.1

Cancel the common factor.

$\frac{0.5}{1} \sec (0)$

Step 9.4.2

Rewrite the expression.

$\frac{\sec (0)}{2}$

Step 9.5

The exact value of $\sec (0)$ is $1$ .

$\frac{1}{2}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$