Calculus Examples

$lim_{x \to \infty} \frac{5 x - 3}{2 x^{2} + 2 x + 4}$

Step 1

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

$lim_{x \to \infty} \frac{\frac{5 x}{x^{2}} + \frac{- 3}{x^{2}}}{\frac{2 x^{2}}{x^{2}} + \frac{2 x}{x^{2}} + \frac{4}{x^{2}}}$

Step 2

Evaluate the limit.

Tap for more steps...

Step 2.1

Simplify each term.

Tap for more steps...

Step 2.1.1

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

Tap for more steps...

Step 2.1.1.1

Factor $x$ out of $5 x$ .

$lim_{x \to \infty} \frac{\frac{x \cdot 5}{x^{2}} + \frac{- 3}{x^{2}}}{\frac{2 x^{2}}{x^{2}} + \frac{2 x}{x^{2}} + \frac{4}{x^{2}}}$

Step 2.1.1.2

Cancel the common factors.

Tap for more steps...

Step 2.1.1.2.1

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

$lim_{x \to \infty} \frac{\frac{x \cdot 5}{x \cdot x} + \frac{- 3}{x^{2}}}{\frac{2 x^{2}}{x^{2}} + \frac{2 x}{x^{2}} + \frac{4}{x^{2}}}$

Step 2.1.1.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{x \cdot 5}{x \cdot x} + \frac{- 3}{x^{2}}}{\frac{2 x^{2}}{x^{2}} + \frac{2 x}{x^{2}} + \frac{4}{x^{2}}}$

Step 2.1.1.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{\frac{5}{x} + \frac{- 3}{x^{2}}}{\frac{2 x^{2}}{x^{2}} + \frac{2 x}{x^{2}} + \frac{4}{x^{2}}}$

Step 2.1.2

Move the negative in front of the fraction.

$lim_{x \to \infty} \frac{\frac{5}{x} - \frac{3}{x^{2}}}{\frac{2 x^{2}}{x^{2}} + \frac{2 x}{x^{2}} + \frac{4}{x^{2}}}$

Step 2.2

Simplify each term.

Tap for more steps...

Step 2.2.1

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

Tap for more steps...

Step 2.2.1.1

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{5}{x} - \frac{3}{x^{2}}}{\frac{2 x^{2}}{x^{2}} + \frac{2 x}{x^{2}} + \frac{4}{x^{2}}}$

Step 2.2.1.2

Divide $2$ by $1$ .

$lim_{x \to \infty} \frac{\frac{5}{x} - \frac{3}{x^{2}}}{2 + \frac{2 x}{x^{2}} + \frac{4}{x^{2}}}$

Step 2.2.2

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

Tap for more steps...

Step 2.2.2.1

Factor $x$ out of $2 x$ .

$lim_{x \to \infty} \frac{\frac{5}{x} - \frac{3}{x^{2}}}{2 + \frac{x \cdot 2}{x^{2}} + \frac{4}{x^{2}}}$

Step 2.2.2.2

Cancel the common factors.

Tap for more steps...

Step 2.2.2.2.1

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

$lim_{x \to \infty} \frac{\frac{5}{x} - \frac{3}{x^{2}}}{2 + \frac{x \cdot 2}{x \cdot x} + \frac{4}{x^{2}}}$

Step 2.2.2.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{5}{x} - \frac{3}{x^{2}}}{2 + \frac{x \cdot 2}{x \cdot x} + \frac{4}{x^{2}}}$

Step 2.2.2.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{\frac{5}{x} - \frac{3}{x^{2}}}{2 + \frac{2}{x} + \frac{4}{x^{2}}}$

Step 2.3

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

$\frac{lim_{x \to \infty} \frac{5}{x} - \frac{3}{x^{2}}}{lim_{x \to \infty} 2 + \frac{2}{x} + \frac{4}{x^{2}}}$

Step 2.4

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

$\frac{lim_{x \to \infty} \frac{5}{x} - lim_{x \to \infty} \frac{3}{x^{2}}}{lim_{x \to \infty} 2 + \frac{2}{x} + \frac{4}{x^{2}}}$

Step 2.5

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{5 lim_{x \to \infty} \frac{1}{x} - lim_{x \to \infty} \frac{3}{x^{2}}}{lim_{x \to \infty} 2 + \frac{2}{x} + \frac{4}{x^{2}}}$

Step 3

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

$\frac{5 \cdot 0 - lim_{x \to \infty} \frac{3}{x^{2}}}{lim_{x \to \infty} 2 + \frac{2}{x} + \frac{4}{x^{2}}}$

Step 4

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{5 \cdot 0 - 3 lim_{x \to \infty} \frac{1}{x^{2}}}{lim_{x \to \infty} 2 + \frac{2}{x} + \frac{4}{x^{2}}}$

Step 5

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

$\frac{5 \cdot 0 - 3 \cdot 0}{lim_{x \to \infty} 2 + \frac{2}{x} + \frac{4}{x^{2}}}$

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 $\infty$ .

$\frac{5 \cdot 0 - 3 \cdot 0}{lim_{x \to \infty} 2 + lim_{x \to \infty} \frac{2}{x} + lim_{x \to \infty} \frac{4}{x^{2}}}$

Step 6.2

Evaluate the limit of $2$ which is constant as $x$ approaches $\infty$ .

$\frac{5 \cdot 0 - 3 \cdot 0}{2 + lim_{x \to \infty} \frac{2}{x} + lim_{x \to \infty} \frac{4}{x^{2}}}$

Step 6.3

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{5 \cdot 0 - 3 \cdot 0}{2 + 2 lim_{x \to \infty} \frac{1}{x} + lim_{x \to \infty} \frac{4}{x^{2}}}$

Step 7

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

$\frac{5 \cdot 0 - 3 \cdot 0}{2 + 2 \cdot 0 + lim_{x \to \infty} \frac{4}{x^{2}}}$

Step 8

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

$\frac{5 \cdot 0 - 3 \cdot 0}{2 + 2 \cdot 0 + 4 lim_{x \to \infty} \frac{1}{x^{2}}}$

Step 9

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

$\frac{5 \cdot 0 - 3 \cdot 0}{2 + 2 \cdot 0 + 4 \cdot 0}$

Step 10

Simplify the answer.

Tap for more steps...

Step 10.1

Simplify the numerator.

Tap for more steps...

Step 10.1.1

Multiply $5$ by $0$ .

$\frac{0 - 3 \cdot 0}{2 + 2 \cdot 0 + 4 \cdot 0}$

Step 10.1.2

Multiply $- 3$ by $0$ .

$\frac{0 + 0}{2 + 2 \cdot 0 + 4 \cdot 0}$

Step 10.1.3

Add $0$ and $0$ .

$\frac{0}{2 + 2 \cdot 0 + 4 \cdot 0}$

Step 10.2

Simplify the denominator.

Tap for more steps...

Step 10.2.1

Multiply $2$ by $0$ .

$\frac{0}{2 + 0 + 4 \cdot 0}$

Step 10.2.2

Multiply $4$ by $0$ .

$\frac{0}{2 + 0 + 0}$

Step 10.2.3

Add $2 + 0$ and $0$ .

$\frac{0}{2 + 0}$

Step 10.2.4

Add $2$ and $0$ .

$\frac{0}{2}$

Step 10.3

Divide $0$ by $2$ .

$0$