Calculus Examples

$lim_{x \to 3} \frac{v (x^{2} - x - 2)}{\sqrt{x - 1}}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $3$ .

$\frac{lim_{x \to 3} v (x^{2} - x - 2)}{lim_{x \to 3} \sqrt{x - 1}}$

Step 2

Move the limit under the radical sign.

$\frac{lim_{x \to 3} v (x^{2} - x - 2)}{\sqrt{lim_{x \to 3} x - 1}}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

$\frac{lim_{x \to 3} v (x^{2} - x - 2)}{\sqrt{lim_{x \to 3} x - lim_{x \to 3} 1}}$

Step 4

Evaluate the limit of $1$ which is constant as $x$ approaches $3$ .

$\frac{lim_{x \to 3} v (x^{2} - x - 2)}{\sqrt{lim_{x \to 3} x - 1 \cdot 1}}$

Step 5

Evaluate the limits by plugging in $3$ for all occurrences of $x$ .

Tap for more steps...

Step 5.1

Evaluate the limit of $v (x^{2} - x - 2)$ by plugging in $3$ for $x$ .

$\frac{v (3^{2} - 1 \cdot 3 - 2)}{\sqrt{lim_{x \to 3} x - 1 \cdot 1}}$

Step 5.2

Simplify each term.

Tap for more steps...

Step 5.2.1

Raise $3$ to the power of $2$ .

$\frac{v (9 - 1 \cdot 3 - 2)}{\sqrt{lim_{x \to 3} x - 1 \cdot 1}}$

Step 5.2.2

Multiply $- 1$ by $3$ .

$\frac{v (9 - 3 - 2)}{\sqrt{lim_{x \to 3} x - 1 \cdot 1}}$

Step 5.3

Subtract $3$ from $9$ .

$\frac{v (6 - 2)}{\sqrt{lim_{x \to 3} x - 1 \cdot 1}}$

Step 5.4

Subtract $2$ from $6$ .

$\frac{v (4)}{\sqrt{lim_{x \to 3} x - 1 \cdot 1}}$

Step 5.5

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{v (4)}{\sqrt{3 - 1 \cdot 1}}$

Step 6

Simplify the answer.

Tap for more steps...

Step 6.1

Simplify the denominator.

Tap for more steps...

Step 6.1.1

Multiply $- 1$ by $1$ .

$\frac{v (4)}{\sqrt{3 - 1}}$

Step 6.1.2

Subtract $1$ from $3$ .

$\frac{v (4)}{\sqrt{2}}$

Step 6.2

Multiply $\frac{v (4)}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{v (4)}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 6.3

Combine and simplify the denominator.

Tap for more steps...

Step 6.3.1

Multiply $\frac{v (4)}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{v (4) \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 6.3.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{v (4) \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 6.3.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{v (4) \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 6.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{v (4) \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 6.3.5

Add $1$ and $1$ .

$\frac{v (4) \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 6.3.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps...

Step 6.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{v (4) \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 6.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{v (4) \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 6.3.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{v (4) \sqrt{2}}{2^{\frac{2}{2}}}$

Step 6.3.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.3.6.4.1

Cancel the common factor.

$\frac{v (4) \sqrt{2}}{2^{\frac{2}{2}}}$

Step 6.3.6.4.2

Rewrite the expression.

$\frac{v (4) \sqrt{2}}{2^{1}}$

Step 6.3.6.5

Evaluate the exponent.

$\frac{v (4) \sqrt{2}}{2}$