Calculus Examples

$\frac{lim_{x \to 8} \sqrt{4 x^{2} - 3 x + 5}}{1 + 9 x}$

Step 1

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 8} 4 x^{2} - 3 x + 5}}{1 + 9 x}$

Step 2

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

$\frac{\sqrt{lim_{x \to 8} 4 x^{2} - lim_{x \to 8} 3 x + lim_{x \to 8} 5}}{1 + 9 x}$

Step 3

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sqrt{4 lim_{x \to 8} x^{2} - lim_{x \to 8} 3 x + lim_{x \to 8} 5}}{1 + 9 x}$

Step 4

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

$\frac{\sqrt{4 {(lim_{x \to 8} x)}^{2} - lim_{x \to 8} 3 x + lim_{x \to 8} 5}}{1 + 9 x}$

Step 5

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sqrt{4 {(lim_{x \to 8} x)}^{2} - 3 lim_{x \to 8} x + lim_{x \to 8} 5}}{1 + 9 x}$

Step 6

Evaluate the limit of $5$ which is constant as $x$ approaches $8$ .

$\frac{\sqrt{4 {(lim_{x \to 8} x)}^{2} - 3 lim_{x \to 8} x + 5}}{1 + 9 x}$

Step 7

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

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Step 7.1

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

$\frac{\sqrt{4 \cdot 8^{2} - 3 lim_{x \to 8} x + 5}}{1 + 9 x}$

Step 7.2

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

$\frac{\sqrt{4 \cdot 8^{2} - 3 \cdot 8 + 5}}{1 + 9 x}$

Step 8

Simplify the numerator.

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Step 8.1

Raise $8$ to the power of $2$ .

$\frac{\sqrt{4 \cdot 64 - 3 \cdot 8 + 5}}{1 + 9 x}$

Step 8.2

Multiply $4$ by $64$ .

$\frac{\sqrt{256 - 3 \cdot 8 + 5}}{1 + 9 x}$

Step 8.3

Multiply $- 3$ by $8$ .

$\frac{\sqrt{256 - 24 + 5}}{1 + 9 x}$

Step 8.4

Subtract $24$ from $256$ .

$\frac{\sqrt{232 + 5}}{1 + 9 x}$

Step 8.5

Add $232$ and $5$ .

$\frac{\sqrt{237}}{1 + 9 x}$