Calculus Examples

$y = \frac{5 x \tan (x)}{x^{2} + 4}$

Step 1

Find the domain to determine if the expression is continuous.

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Step 1.1

Set the denominator in $\frac{5 x \tan (x)}{x^{2} + 4}$ equal to $0$ to find where the expression is undefined.

$x^{2} + 4 = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Subtract $4$ from both sides of the equation.

$x^{2} = - 4$

Step 1.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 4}$

Step 1.2.3

Simplify $\pm \sqrt{- 4}$ .

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Step 1.2.3.1

Rewrite $- 4$ as $- 1 (4)$ .

$x = \pm \sqrt{- 1 (4)}$

Step 1.2.3.2

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{4}$

Step 1.2.3.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \cdot \sqrt{4}$

Step 1.2.3.4

Rewrite $4$ as $2^{2}$ .

$x = \pm i \cdot \sqrt{2^{2}}$

Step 1.2.3.5

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm i \cdot 2$

Step 1.2.3.6

Move $2$ to the left of $i$ .

$x = \pm 2 i$

Step 1.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 2 i$

Step 1.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 2 i$

Step 1.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2 i, - 2 i$

Step 1.3

Set the argument in $\tan (x)$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

$x = \frac{π}{2} + π n$ , for any integer $n$

Step 1.4

The domain is all values of $x$ that make the expression defined.

Set-Builder Notation:

${x | x \neq \frac{π}{2} + π n}$ , for any integer $n$

Set-Builder Notation:

${x | x \neq \frac{π}{2} + π n}$ , for any integer $n$

Step 2

Since the domain is not all real numbers, $\frac{5 x \tan (x)}{x^{2} + 4}$ is not continuous over all real numbers.

Not continuous

Step 3