Calculus Examples

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

Step 1

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

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

Step 2

Move the limit under the radical sign.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Evaluate the limit of $9$ which is constant as $x$ approaches $8$ .

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

Step 7

Move the limit under the radical sign.

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

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

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Step 12.1

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

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

Step 12.2

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

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

Step 13

Simplify each term.

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Step 13.1

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

$\sqrt{4 \cdot 64 + 9} - \sqrt{4 \cdot 8^{2} - 1 \cdot 5}$

Step 13.2

Multiply $4$ by $64$ .

$\sqrt{256 + 9} - \sqrt{4 \cdot 8^{2} - 1 \cdot 5}$

Step 13.3

Add $256$ and $9$ .

$\sqrt{265} - \sqrt{4 \cdot 8^{2} - 1 \cdot 5}$

Step 13.4

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

$\sqrt{265} - \sqrt{4 \cdot 64 - 1 \cdot 5}$

Step 13.5

Multiply $4$ by $64$ .

$\sqrt{265} - \sqrt{256 - 1 \cdot 5}$

Step 13.6

Multiply $- 1$ by $5$ .

$\sqrt{265} - \sqrt{256 - 5}$

Step 13.7

Subtract $5$ from $256$ .

$\sqrt{265} - \sqrt{251}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\sqrt{265} - \sqrt{251}$

Decimal Form:

$0.43584107 \dots$