Calculus Examples

$lim_{x \to 8.999} \frac{\sqrt{x} - 3}{x - 9}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $8.999$ .

$\frac{lim_{x \to 8.999} \sqrt{x} - 3}{lim_{x \to 8.999} x - 9}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8.999$ .

$\frac{lim_{x \to 8.999} \sqrt{x} - lim_{x \to 8.999} 3}{lim_{x \to 8.999} x - 9}$

Step 3

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 8.999} x} - lim_{x \to 8.999} 3}{lim_{x \to 8.999} x - 9}$

Step 4

Evaluate the limit of $3$ which is constant as $x$ approaches $8.999$ .

$\frac{\sqrt{lim_{x \to 8.999} x} - 1 \cdot 3}{lim_{x \to 8.999} x - 9}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8.999$ .

$\frac{\sqrt{lim_{x \to 8.999} x} - 1 \cdot 3}{lim_{x \to 8.999} x - lim_{x \to 8.999} 9}$

Step 6

Evaluate the limit of $9$ which is constant as $x$ approaches $8.999$ .

$\frac{\sqrt{lim_{x \to 8.999} x} - 1 \cdot 3}{lim_{x \to 8.999} x - 1 \cdot 9}$

Step 7

Evaluate the limits by plugging in $8.999$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $8.999$ for $x$ .

$\frac{\sqrt{8.999} - 1 \cdot 3}{lim_{x \to 8.999} x - 1 \cdot 9}$

Step 7.2

Evaluate the limit of $x$ by plugging in $8.999$ for $x$ .

$\frac{\sqrt{8.999} - 1 \cdot 3}{8.999 - 1 \cdot 9}$

Step 8

Simplify the answer.

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Step 8.1

Multiply $- 1$ by $3$ .

$\frac{\sqrt{8.999} - 3}{8.999 - 1 \cdot 9}$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Multiply $- 1$ by $9$ .

$\frac{\sqrt{8.999} - 3}{8.999 - 9}$

Step 8.2.2

Subtract $9$ from $8.999$ .

$\frac{\sqrt{8.999} - 3}{- 0.001}$

Step 8.3

Move the negative in front of the fraction.

$- \frac{\sqrt{8.999} - 3}{0.001}$

Step 8.4

Evaluate the root.

$- \frac{2.99983332 - 3}{0.001}$

Step 8.5

Subtract $3$ from $2.99983332$ .

$- \frac{- 0.00016667}{0.001}$

Step 8.6

Divide $- 0.00016667$ by $0.001$ .

$- - 0.16667129$

Step 8.7

Multiply $- 1$ by $- 0.16667129$ .

$0.16667129$