Calculus Examples

$lim_{x \to 8} (x - 1) \cdot \frac{x - \sqrt{x^{2} - 6 x}}{x + 2}$

Step 1

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

$lim_{x \to 8} x - 1 \cdot lim_{x \to 8} \frac{x - \sqrt{x^{2} - 6 x}}{x + 2}$

Step 2

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

$(lim_{x \to 8} x - lim_{x \to 8} 1) \cdot lim_{x \to 8} \frac{x - \sqrt{x^{2} - 6 x}}{x + 2}$

Step 3

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

$(lim_{x \to 8} x - 1 \cdot 1) \cdot lim_{x \to 8} \frac{x - \sqrt{x^{2} - 6 x}}{x + 2}$

Step 4

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

$(lim_{x \to 8} x - 1 \cdot 1) \cdot \frac{lim_{x \to 8} x - \sqrt{x^{2} - 6 x}}{lim_{x \to 8} x + 2}$

Step 5

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

$(lim_{x \to 8} x - 1 \cdot 1) \cdot \frac{lim_{x \to 8} x - lim_{x \to 8} \sqrt{x^{2} - 6 x}}{lim_{x \to 8} x + 2}$

Step 6

Move the limit under the radical sign.

$(lim_{x \to 8} x - 1 \cdot 1) \cdot \frac{lim_{x \to 8} x - \sqrt{lim_{x \to 8} x^{2} - 6 x}}{lim_{x \to 8} x + 2}$

Step 7

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

$(lim_{x \to 8} x - 1 \cdot 1) \cdot \frac{lim_{x \to 8} x - \sqrt{lim_{x \to 8} x^{2} - lim_{x \to 8} 6 x}}{lim_{x \to 8} x + 2}$

Step 8

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

$(lim_{x \to 8} x - 1 \cdot 1) \cdot \frac{lim_{x \to 8} x - \sqrt{{(lim_{x \to 8} x)}^{2} - lim_{x \to 8} 6 x}}{lim_{x \to 8} x + 2}$

Step 9

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$(lim_{x \to 8} x - 1 \cdot 1) \cdot \frac{lim_{x \to 8} x - \sqrt{{(lim_{x \to 8} x)}^{2} - 6 lim_{x \to 8} x}}{lim_{x \to 8} x + 2}$

Step 10

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

$(lim_{x \to 8} x - 1 \cdot 1) \cdot \frac{lim_{x \to 8} x - \sqrt{{(lim_{x \to 8} x)}^{2} - 6 lim_{x \to 8} x}}{lim_{x \to 8} x + lim_{x \to 8} 2}$

Step 11

Evaluate the limit of $2$ which is constant as $x$ approaches $8$ .

$(lim_{x \to 8} x - 1 \cdot 1) \cdot \frac{lim_{x \to 8} x - \sqrt{{(lim_{x \to 8} x)}^{2} - 6 lim_{x \to 8} x}}{lim_{x \to 8} x + 2}$

Step 12

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

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Step 12.1