Calculus Examples

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

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 $0$ .

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

Step 1.2

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

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

Step 1.3

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

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

Step 2

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

$\frac{lim_{x \to 0} x \cot (x)}{\sec (2 \cdot 0)}$

Step 3

Consider the left sided limit.

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

Step 4

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

$\begin{array}{c} x & x \cot (x) \\ - 0.1 & 0.99666444 \\ - 0.01 & 0.99996666 \\ - 0.001 & 0.99999966 \end{array}$

Step 5

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

$1$

Step 6

Consider the right sided limit.

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

Step 7

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

$\begin{array}{c} x & x \cot (x) \\ 0.1 & 0.99666444 \\ 0.01 & 0.99996666 \\ 0.001 & 0.99999966 \end{array}$

Step 8

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

$\frac{1}{\sec (2 \cdot 0)}$

Step 9

Simplify the answer.

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Step 9.1

Rewrite $\sec (2 \cdot 0)$ in terms of sines and cosines.

$\frac{1}{\frac{1}{\cos (2 \cdot 0)}}$

Step 9.2

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (2 \cdot 0)}$ .

$1 \cos (2 \cdot 0)$

Step 9.3

Multiply $\cos (2 \cdot 0)$ by $1$ .

$\cos (2 \cdot 0)$

Step 9.4

Multiply $2$ by $0$ .

$\cos (0)$

Step 9.5

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

$1$