Calculus Examples

$\frac{lim_{x \to 8} \sqrt{7 + 8 x^{2}}}{10 + 7 x}$

Step 1

Evaluate the limit.

Tap for more steps...

Step 1.1

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 8} 7 + 8 x^{2}}}{10 + 7 x}$

Step 1.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{\sqrt{lim_{x \to 8} 7 + lim_{x \to 8} 8 x^{2}}}{10 + 7 x}$

Step 1.3

Evaluate the limit of $7$ which is constant as $x$ approaches $8$ .

$\frac{\sqrt{7 + lim_{x \to 8} 8 x^{2}}}{10 + 7 x}$

Step 1.4

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sqrt{7 + 8 lim_{x \to 8} x^{2}}}{10 + 7 x}$

Step 1.5

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{\sqrt{7 + 8 {(lim_{x \to 8} x)}^{2}}}{10 + 7 x}$

Step 2

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{\sqrt{7 + 8 \cdot 8^{2}}}{10 + 7 x}$

Step 3

Simplify the numerator.

Tap for more steps...

Step 3.1

Multiply $8$ by $8^{2}$ by adding the exponents.

Tap for more steps...

Step 3.1.1

Multiply $8$ by $8^{2}$ .

Tap for more steps...

Step 3.1.1.1

Raise $8$ to the power of $1$ .

$\frac{\sqrt{7 + 8^{1} \cdot 8^{2}}}{10 + 7 x}$

Step 3.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt{7 + 8^{1 + 2}}}{10 + 7 x}$

Step 3.1.2

Add $1$ and $2$ .

$\frac{\sqrt{7 + 8^{3}}}{10 + 7 x}$

Step 3.2

Raise $8$ to the power of $3$ .

$\frac{\sqrt{7 + 512}}{10 + 7 x}$

Step 3.3

Add $7$ and $512$ .

$\frac{\sqrt{519}}{10 + 7 x}$