Calculus Examples

$f (x) = \frac{2 x^{2} + 5 x - 3}{x + 3}$

Step 1

Find the domain to determine if the expression is continuous.

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Step 1.1

Set the denominator in $\frac{2 x^{2} + 5 x - 3}{x + 3}$ equal to $0$ to find where the expression is undefined.

$x + 3 = 0$

Step 1.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 1.3

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

Interval Notation:

$(- \infty, - 3) \cup (- 3, \infty)$

Set-Builder Notation:

${x | x \neq - 3}$

Interval Notation:

$(- \infty, - 3) \cup (- 3, \infty)$

Set-Builder Notation:

${x | x \neq - 3}$

Step 2

Since the domain is not all real numbers, $\frac{2 x^{2} + 5 x - 3}{x + 3}$ is not continuous over all real numbers.

Not continuous

Step 3

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$