Calculus Examples

$lim_{x \to 81} \frac{x - 81}{\sqrt{x - 9}}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $81$ .

$\frac{lim_{x \to 81} x - 81}{lim_{x \to 81} \sqrt{x - 9}}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $81$ .

$\frac{lim_{x \to 81} x - lim_{x \to 81} 81}{lim_{x \to 81} \sqrt{x - 9}}$

Step 3

Evaluate the limit of $81$ which is constant as $x$ approaches $81$ .

$\frac{lim_{x \to 81} x - 1 \cdot 81}{lim_{x \to 81} \sqrt{x - 9}}$

Step 4

Move the limit under the radical sign.

$\frac{lim_{x \to 81} x - 1 \cdot 81}{\sqrt{lim_{x \to 81} x - 9}}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $81$ .

$\frac{lim_{x \to 81} x - 1 \cdot 81}{\sqrt{lim_{x \to 81} x - lim_{x \to 81} 9}}$

Step 6

Evaluate the limit of $9$ which is constant as $x$ approaches $81$ .

$\frac{lim_{x \to 81} x - 1 \cdot 81}{\sqrt{lim_{x \to 81} x - 1 \cdot 9}}$

Step 7

Evaluate the limits by plugging in $81$ for all occurrences of $x$ .

Tap for more steps...

Step 7.1

Evaluate the limit of $x$ by plugging in $81$ for $x$ .

$\frac{81 - 1 \cdot 81}{\sqrt{lim_{x \to 81} x - 1 \cdot 9}}$

Step 7.2

Evaluate the limit of $x$ by plugging in $81$ for $x$ .

$\frac{81 - 1 \cdot 81}{\sqrt{81 - 1 \cdot 9}}$

Step 8

Simplify the answer.

Tap for more steps...

Step 8.1

Simplify the numerator.

Tap for more steps...

Step 8.1.1

Multiply $- 1$ by $81$ .

$\frac{81 - 81}{\sqrt{81 - 1 \cdot 9}}$

Step 8.1.2

Subtract $81$ from $81$ .

$\frac{0}{\sqrt{81 - 1 \cdot 9}}$

Step 8.2

Simplify the denominator.

Tap for more steps...

Step 8.2.1

Multiply $- 1$ by $9$ .

$\frac{0}{\sqrt{81 - 9}}$

Step 8.2.2

Subtract $9$ from $81$ .

$\frac{0}{\sqrt{72}}$

Step 8.2.3

Rewrite $72$ as $6^{2} \cdot 2$ .

Tap for more steps...

Step 8.2.3.1

Factor $36$ out of $72$ .

$\frac{0}{\sqrt{36 (2)}}$

Step 8.2.3.2

Rewrite $36$ as $6^{2}$ .

$\frac{0}{\sqrt{6^{2} \cdot 2}}$

Step 8.2.4

Pull terms out from under the radical.

$\frac{0}{6 \sqrt{2}}$

Step 8.3

Cancel the common factor of $0$ and $6$ .

Tap for more steps...

Step 8.3.1

Factor $6$ out of $0$ .

$\frac{6 (0)}{6 \sqrt{2}}$

Step 8.3.2

Cancel the common factors.

Tap for more steps...

Step 8.3.2.1

Factor $6$ out of $6 \sqrt{2}$ .

$\frac{6 (0)}{6 (\sqrt{2})}$

Step 8.3.2.2

Cancel the common factor.

$\frac{6 \cdot 0}{6 \sqrt{2}}$

Step 8.3.2.3

Rewrite the expression.

$\frac{0}{\sqrt{2}}$

Step 8.4

Divide $0$ by $\sqrt{2}$ .

$0$