Calculus Examples

$lim_{x \to \infty} \sqrt{x^{2} + 8 x + 1} - x$

Step 1

Multiply to rationalize the numerator.

$lim_{x \to \infty} \frac{(\sqrt{x^{2} + 8 x + 1} - x) (\sqrt{x^{2} + 8 x + 1} + x)}{\sqrt{x^{2} + 8 x + 1} + x}$

Step 2

Simplify.

Tap for more steps...

Step 2.1

Expand the numerator using the FOIL method.

$lim_{x \to \infty} \frac{{\sqrt{x^{2} + 8 x + 1}}^{2} + \sqrt{x^{2} + 8 x + 1} x + \sqrt{x^{2} + 8 x + 1} (- x) - x^{2}}{\sqrt{x^{2} + 8 x + 1} + x}$

Step 2.2

Simplify.

Tap for more steps...

Step 2.2.1

Subtract $x^{2}$ from $x^{2}$ .

$lim_{x \to \infty} \frac{8 x + 1 + 0}{\sqrt{x^{2} + 8 x + 1} + x}$

Step 2.2.2

Add $8 x + 1$ and $0$ .

$lim_{x \to \infty} \frac{8 x + 1}{\sqrt{x^{2} + 8 x + 1} + x}$

Step 3

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x = \sqrt{x^{2}}$ .

$lim_{x \to \infty} \frac{\frac{8 x}{x} + \frac{1}{x}}{\sqrt{\frac{x^{2}}{x^{2}} + \frac{8 x}{x^{2}} + \frac{1}{x^{2}}} + \frac{x}{x}}$

Step 4

Evaluate the limit.

Tap for more steps...

Step 4.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 4.1.1

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{8 x}{x} + \frac{1}{x}}{\sqrt{\frac{x^{2}}{x^{2}} + \frac{8 x}{x^{2}} + \frac{1}{x^{2}}} + \frac{x}{x}}$

Step 4.1.2

Divide $8$ by $1$ .

$lim_{x \to \infty} \frac{8 + \frac{1}{x}}{\sqrt{\frac{x^{2}}{x^{2}} + \frac{8 x}{x^{2}} + \frac{1}{x^{2}}} + \frac{x}{x}}$

Step 4.2

Simplify terms.

Tap for more steps...

Step 4.2.1

Cancel the common factor of $x$ .

$lim_{x \to \infty} \frac{8 + \frac{1}{x}}{\sqrt{\frac{x^{2}}{x^{2}} + \frac{8 x}{x^{2}} + \frac{1}{x^{2}}} + 1}$

Step 4.2.2

Simplify each term.

Tap for more steps...

Step 4.2.2.1

Cancel the common factor of $x^{2}$ .

Tap for more steps...

Step 4.2.2.1.1

Cancel the common factor.

$lim_{x \to \infty} \frac{8 + \frac{1}{x}}{\sqrt{\frac{x^{2}}{x^{2}} + \frac{8 x}{x^{2}} + \frac{1}{x^{2}}} + 1}$

Step 4.2.2.1.2

Rewrite the expression.

$lim_{x \to \infty} \frac{8 + \frac{1}{x}}{\sqrt{1 + \frac{8 x}{x^{2}} + \frac{1}{x^{2}}} + 1}$

Step 4.2.2.2

Cancel the common factor of $x$ and $x^{2}$ .

Tap for more steps...

Step 4.2.2.2.1

Factor $x$ out of $8 x$ .

$lim_{x \to \infty} \frac{8 + \frac{1}{x}}{\sqrt{1 + \frac{x \cdot 8}{x^{2}} + \frac{1}{x^{2}}} + 1}$

Step 4.2.2.2.2

Cancel the common factors.

Tap for more steps...

Step 4.2.2.2.2.1

Factor $x$ out of $x^{2}$ .

$lim_{x \to \infty} \frac{8 + \frac{1}{x}}{\sqrt{1 + \frac{x \cdot 8}{x \cdot x} + \frac{1}{x^{2}}} + 1}$

Step 4.2.2.2.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{8 + \frac{1}{x}}{\sqrt{1 + \frac{x \cdot 8}{x \cdot x} + \frac{1}{x^{2}}} + 1}$

Step 4.2.2.2.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{8 + \frac{1}{x}}{\sqrt{1 + \frac{8}{x} + \frac{1}{x^{2}}} + 1}$

Step 4.3

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

$\frac{lim_{x \to \infty} 8 + \frac{1}{x}}{lim_{x \to \infty} \sqrt{1 + \frac{8}{x} + \frac{1}{x^{2}}} + 1}$

Step 4.4

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

$\frac{lim_{x \to \infty} 8 + lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} \sqrt{1 + \frac{8}{x} + \frac{1}{x^{2}}} + 1}$

Step 4.5

Evaluate the limit of $8$ which is constant as $x$ approaches $\infty$ .

$\frac{8 + lim_{x \to \infty} \frac{1}{x}}{lim_{x \to \infty} \sqrt{1 + \frac{8}{x} + \frac{1}{x^{2}}} + 1}$

Step 5

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x}$ approaches $0$ .

$\frac{8 + 0}{lim_{x \to \infty} \sqrt{1 + \frac{8}{x} + \frac{1}{x^{2}}} + 1}$

Step 6

Evaluate the limit.

Tap for more steps...

Step 6.1

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

$\frac{8 + 0}{lim_{x \to \infty} \sqrt{1 + \frac{8}{x} + \frac{1}{x^{2}}} + lim_{x \to \infty} 1}$

Step 6.2

Move the limit under the radical sign.

$\frac{8 + 0}{\sqrt{lim_{x \to \infty} 1 + \frac{8}{x} + \frac{1}{x^{2}}} + lim_{x \to \infty} 1}$

Step 6.3

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

$\frac{8 + 0}{\sqrt{lim_{x \to \infty} 1 + lim_{x \to \infty} \frac{8}{x} + lim_{x \to \infty} \frac{1}{x^{2}}} + lim_{x \to \infty} 1}$

Step 6.4

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

$\frac{8 + 0}{\sqrt{1 + lim_{x \to \infty} \frac{8}{x} + lim_{x \to \infty} \frac{1}{x^{2}}} + lim_{x \to \infty} 1}$

Step 6.5

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$\frac{8 + 0}{\sqrt{1 + 8 lim_{x \to \infty} \frac{1}{x} + lim_{x \to \infty} \frac{1}{x^{2}}} + lim_{x \to \infty} 1}$

Step 7

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x}$ approaches $0$ .

$\frac{8 + 0}{\sqrt{1 + 8 \cdot 0 + lim_{x \to \infty} \frac{1}{x^{2}}} + lim_{x \to \infty} 1}$

Step 8

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{2}}$ approaches $0$ .

$\frac{8 + 0}{\sqrt{1 + 8 \cdot 0 + 0} + lim_{x \to \infty} 1}$

Step 9

Evaluate the limit.

Tap for more steps...

Step 9.1

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

$\frac{8 + 0}{\sqrt{1 + 8 \cdot 0 + 0} + 1}$

Step 9.2

Simplify the answer.

Tap for more steps...

Step 9.2.1

Add $8$ and $0$ .

$\frac{8}{\sqrt{1 + 8 \cdot 0 + 0} + 1}$

Step 9.2.2

Simplify the denominator.

Tap for more steps...

Step 9.2.2.1

Multiply $8$ by $0$ .

$\frac{8}{\sqrt{1 + 0 + 0} + 1}$

Step 9.2.2.2

Add $1 + 0$ and $0$ .

$\frac{8}{\sqrt{1 + 0} + 1}$

Step 9.2.2.3

Add $1$ and $0$ .

$\frac{8}{\sqrt{1} + 1}$

Step 9.2.2.4

Any root of $1$ is $1$ .

$\frac{8}{1 + 1}$

Step 9.2.2.5

Add $1$ and $1$ .

$\frac{8}{2}$

Step 9.2.3

Divide $8$ by $2$ .

$4$