Calculus Examples

$lim_{x \to \frac{2}{3}} \sqrt{9 x + 19}$

Step 1

Evaluate the limit.

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Step 1.1

Move the limit under the radical sign.

$\sqrt{lim_{x \to \frac{2}{3}} 9 x + 19}$

Step 1.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{2}{3}$ .

$\sqrt{lim_{x \to \frac{2}{3}} 9 x + lim_{x \to \frac{2}{3}} 19}$

Step 1.3

Move the term $9$ outside of the limit because it is constant with respect to $x$ .

$\sqrt{9 lim_{x \to \frac{2}{3}} x + lim_{x \to \frac{2}{3}} 19}$

Step 1.4

Evaluate the limit of $19$ which is constant as $x$ approaches $\frac{2}{3}$ .

$\sqrt{9 lim_{x \to \frac{2}{3}} x + 19}$

Step 2

Evaluate the limit of $x$ by plugging in $\frac{2}{3}$ for $x$ .

$\sqrt{9 (\frac{2}{3}) + 19}$

Step 3

Simplify the answer.

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Step 3.1

Cancel the common factor of $3$ .

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Step 3.1.1

Factor $3$ out of $9$ .

$\sqrt{3 (3) \frac{2}{3} + 19}$

Step 3.1.2

Cancel the common factor.

$\sqrt{3 \cdot 3 \frac{2}{3} + 19}$

Step 3.1.3

Rewrite the expression.

$\sqrt{3 \cdot 2 + 19}$

Step 3.2

Multiply $3$ by $2$ .

$\sqrt{6 + 19}$

Step 3.3

Add $6$ and $19$ .

$\sqrt{25}$

Step 3.4

Rewrite $25$ as $5^{2}$ .

$\sqrt{5^{2}}$

Step 3.5

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

$5$