Calculus Examples

$\frac{lim_{x \to 3} 4 x - 3}{\sqrt{x + 1}}$

Step 1

Evaluate the limit.

Tap for more steps...

Step 1.1

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

$\frac{lim_{x \to 3} 4 x - lim_{x \to 3} 3}{\sqrt{x + 1}}$

Step 1.2

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{4 lim_{x \to 3} x - lim_{x \to 3} 3}{\sqrt{x + 1}}$

Step 1.3

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

$\frac{4 lim_{x \to 3} x - 1 \cdot 3}{\sqrt{x + 1}}$

Step 2

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

$\frac{4 \cdot 3 - 1 \cdot 3}{\sqrt{x + 1}}$

Step 3

Simplify the answer.

Tap for more steps...

Step 3.1

Simplify the numerator.

Tap for more steps...

Step 3.1.1

Multiply $4$ by $3$ .

$\frac{12 - 1 \cdot 3}{\sqrt{x + 1}}$

Step 3.1.2

Multiply $- 1$ by $3$ .

$\frac{12 - 3}{\sqrt{x + 1}}$

Step 3.1.3

Subtract $3$ from $12$ .

$\frac{9}{\sqrt{x + 1}}$

Step 3.2

Multiply $\frac{9}{\sqrt{x + 1}}$ by $\frac{\sqrt{x + 1}}{\sqrt{x + 1}}$ .

$\frac{9}{\sqrt{x + 1}} \cdot \frac{\sqrt{x + 1}}{\sqrt{x + 1}}$

Step 3.3

Combine and simplify the denominator.

Tap for more steps...

Step 3.3.1

Multiply $\frac{9}{\sqrt{x + 1}}$ by $\frac{\sqrt{x + 1}}{\sqrt{x + 1}}$ .

$\frac{9 \sqrt{x + 1}}{\sqrt{x + 1} \sqrt{x + 1}}$

Step 3.3.2

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

$\frac{9 \sqrt{x + 1}}{{\sqrt{x + 1}}^{1} \sqrt{x + 1}}$

Step 3.3.3

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

$\frac{9 \sqrt{x + 1}}{{\sqrt{x + 1}}^{1} {\sqrt{x + 1}}^{1}}$

Step 3.3.4

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

$\frac{9 \sqrt{x + 1}}{{\sqrt{x + 1}}^{1 + 1}}$

Step 3.3.5

Add $1$ and $1$ .

$\frac{9 \sqrt{x + 1}}{{\sqrt{x + 1}}^{2}}$

Step 3.3.6

Rewrite ${\sqrt{x + 1}}^{2}$ as $x + 1$ .

Tap for more steps...

Step 3.3.6.1

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

$\frac{9 \sqrt{x + 1}}{{({(x + 1)}^{\frac{1}{2}})}^{2}}$

Step 3.3.6.2

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

$\frac{9 \sqrt{x + 1}}{{(x + 1)}^{\frac{1}{2} \cdot 2}}$

Step 3.3.6.3

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

$\frac{9 \sqrt{x + 1}}{{(x + 1)}^{\frac{2}{2}}}$

Step 3.3.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.6.4.1

Cancel the common factor.

$\frac{9 \sqrt{x + 1}}{{(x + 1)}^{\frac{2}{2}}}$

Step 3.3.6.4.2

Rewrite the expression.

$\frac{9 \sqrt{x + 1}}{{(x + 1)}^{1}}$

Step 3.3.6.5

Simplify.

$\frac{9 \sqrt{x + 1}}{x + 1}$