Calculus Examples

$g (x) = \frac{2 x^{2} + 5}{2 x - 10}$

Step 1

Find the domain to determine if the expression is continuous.

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Step 1.1

Set the denominator in $\frac{2 x^{2} + 5}{2 x - 10}$ equal to $0$ to find where the expression is undefined.

$2 x - 10 = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Add $10$ to both sides of the equation.

$2 x = 10$

Step 1.2.2

Divide each term in $2 x = 10$ by $2$ and simplify.

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Step 1.2.2.1

Divide each term in $2 x = 10$ by $2$ .

$\frac{2 x}{2} = \frac{10}{2}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{10}{2}$

Step 1.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{10}{2}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $10$ by $2$ .

$x = 5$

Step 1.3

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

Interval Notation:

$(- \infty, 5) \cup (5, \infty)$

Set-Builder Notation:

${x | x \neq 5}$

Interval Notation:

$(- \infty, 5) \cup (5, \infty)$

Set-Builder Notation:

${x | x \neq 5}$

Step 2

Since the domain is not all real numbers, $\frac{2 x^{2} + 5}{2 x - 10}$ is not continuous over all real numbers.

Not continuous

Step 3