Calculus Examples

$\frac{lim_{u \to 2} \sqrt{4 u + 1} - 3}{u - 2}$

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 $u$ approaches $2$ .

$\frac{lim_{u \to 2} \sqrt{4 u + 1} - lim_{u \to 2} 3}{u - 2}$

Step 1.2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{u \to 2} 4 u + 1} - lim_{u \to 2} 3}{u - 2}$

Step 1.3

Split the limit using the Sum of Limits Rule on the limit as $u$ approaches $2$ .

$\frac{\sqrt{lim_{u \to 2} 4 u + lim_{u \to 2} 1} - lim_{u \to 2} 3}{u - 2}$

Step 1.4

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

$\frac{\sqrt{4 lim_{u \to 2} u + lim_{u \to 2} 1} - lim_{u \to 2} 3}{u - 2}$

Step 1.5

Evaluate the limit of $1$ which is constant as $u$ approaches $2$ .

$\frac{\sqrt{4 lim_{u \to 2} u + 1} - lim_{u \to 2} 3}{u - 2}$

Step 1.6

Evaluate the limit of $3$ which is constant as $u$ approaches $2$ .

$\frac{\sqrt{4 lim_{u \to 2} u + 1} - 1 \cdot 3}{u - 2}$

Step 2

Evaluate the limit of $u$ by plugging in $2$ for $u$ .

$\frac{\sqrt{4 \cdot 2 + 1} - 1 \cdot 3}{u - 2}$

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 $2$ .

$\frac{\sqrt{8 + 1} - 1 \cdot 3}{u - 2}$

Step 3.1.2

Add $8$ and $1$ .

$\frac{\sqrt{9} - 1 \cdot 3}{u - 2}$

Step 3.1.3

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{3^{2}} - 1 \cdot 3}{u - 2}$

Step 3.1.4

Pull terms out from under the radical, assuming positive real numbers.

$\frac{3 - 1 \cdot 3}{u - 2}$

Step 3.1.5

Multiply $- 1$ by $3$ .

$\frac{3 - 3}{u - 2}$

Step 3.1.6

Subtract $3$ from $3$ .

$\frac{0}{u - 2}$

Step 3.2

Divide $0$ by $u - 2$ .

$0$