Calculus Examples

$lim_{x \to 8} (- 5 x + 9) \sqrt{68 - x^{2}}$

Step 1

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

$lim_{x \to 8} - 5 x + 9 \cdot lim_{x \to 8} \sqrt{68 - x^{2}}$

Step 2

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

$(- lim_{x \to 8} 5 x + lim_{x \to 8} 9) \cdot lim_{x \to 8} \sqrt{68 - x^{2}}$

Step 3

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$(- 5 lim_{x \to 8} x + lim_{x \to 8} 9) \cdot lim_{x \to 8} \sqrt{68 - x^{2}}$

Step 4

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

$(- 5 lim_{x \to 8} x + 9) \cdot lim_{x \to 8} \sqrt{68 - x^{2}}$

Step 5

Move the limit under the radical sign.

$(- 5 lim_{x \to 8} x + 9) \cdot \sqrt{lim_{x \to 8} 68 - x^{2}}$

Step 6

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

$(- 5 lim_{x \to 8} x + 9) \cdot \sqrt{lim_{x \to 8} 68 - lim_{x \to 8} x^{2}}$

Step 7

Evaluate the limit of $68$ which is constant as $x$ approaches $8$ .

$(- 5 lim_{x \to 8} x + 9) \cdot \sqrt{68 - lim_{x \to 8} x^{2}}$

Step 8

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

$(- 5 lim_{x \to 8} x + 9) \cdot \sqrt{68 - {(lim_{x \to 8} x)}^{2}}$

Step 9

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

Tap for more steps...

Step 9.1

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

$(- 5 \cdot 8 + 9) \cdot \sqrt{68 - {(lim_{x \to 8} x)}^{2}}$

Step 9.2

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

$(- 5 \cdot 8 + 9) \cdot \sqrt{68 - 8^{2}}$

Step 10

Simplify the answer.

Tap for more steps...

Step 10.1

Multiply $- 5$ by $8$ .

$(- 40 + 9) \cdot \sqrt{68 - 8^{2}}$

Step 10.2

Add $- 40$ and $9$ .

$- 31 \cdot \sqrt{68 - 8^{2}}$

Step 10.3

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

$- 31 \cdot \sqrt{68 - 1 \cdot 64}$

Step 10.4

Multiply $- 1$ by $64$ .

$- 31 \cdot \sqrt{68 - 64}$

Step 10.5

Subtract $64$ from $68$ .

$- 31 \cdot \sqrt{4}$

Step 10.6

Rewrite $4$ as $2^{2}$ .

$- 31 \cdot \sqrt{2^{2}}$

Step 10.7

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

$- 31 \cdot 2$

Step 10.8

Multiply $- 31$ by $2$ .

$- 62$