Calculus Examples

$\frac{lim_{h \to 0} \sqrt{8 (x + h) + 9}}{h} - \frac{\sqrt{8 x + 9}}{h}$

Step 1

Evaluate the limit.

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Step 1.1

Move the limit under the radical sign.

$\frac{\sqrt{lim_{h \to 0} 8 (x + h) + 9}}{h} - \frac{\sqrt{8 x + 9}}{h}$

Step 1.2

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$\frac{\sqrt{lim_{h \to 0} 8 (x + h) + lim_{h \to 0} 9}}{h} - \frac{\sqrt{8 x + 9}}{h}$

Step 1.3

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

$\frac{\sqrt{8 lim_{h \to 0} x + h + lim_{h \to 0} 9}}{h} - \frac{\sqrt{8 x + 9}}{h}$

Step 1.4

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$\frac{\sqrt{8 (lim_{h \to 0} x + lim_{h \to 0} h) + lim_{h \to 0} 9}}{h} - \frac{\sqrt{8 x + 9}}{h}$

Step 1.5

Evaluate the limit of $x$ which is constant as $h$ approaches $0$ .

$\frac{\sqrt{8 (x + lim_{h \to 0} h) + lim_{h \to 0} 9}}{h} - \frac{\sqrt{8 x + 9}}{h}$

Step 1.6

Evaluate the limit of $9$ which is constant as $h$ approaches $0$ .

$\frac{\sqrt{8 (x + lim_{h \to 0} h) + 9}}{h} - \frac{\sqrt{8 x + 9}}{h}$

Step 2

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$\frac{\sqrt{8 (x + 0) + 9}}{h} - \frac{\sqrt{8 x + 9}}{h}$

Step 3

Combine the opposite terms in $\frac{\sqrt{8 (x + 0) + 9}}{h} - \frac{\sqrt{8 x + 9}}{h}$ .

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Step 3.1

Add $x$ and $0$ .

$\frac{\sqrt{8 x + 9}}{h} - \frac{\sqrt{8 x + 9}}{h}$

Step 3.2

Subtract $\frac{\sqrt{8 x + 9}}{h}$ from $\frac{\sqrt{8 x + 9}}{h}$ .

$0$