Calculus Examples

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

Step 1

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x^{5}$ .

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

Step 2

Evaluate the limit.

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Step 2.1

Cancel the common factor of $x^{3}$ and $x^{5}$ .

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Step 2.1.1

Multiply by $1$ .

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

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

Factor $x^{3}$ out of $x^{5}$ .

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

Step 2.1.2.2

Cancel the common factor.

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

Step 2.1.2.3

Rewrite the expression.

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

Step 2.2

Cancel the common factor of $x^{5}$ .

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Divide $4$ by $1$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 3

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{5}}$ approaches $0$ .

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

Step 4

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

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

Step 5

Evaluate the limit.

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Step 5.1

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

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

Step 5.2

Evaluate the limit of $4$ which is constant as $x$ approaches $\infty$ .

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

Step 6

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{5}}$ approaches $0$ .

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

Step 7

Simplify the answer.

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Step 7.1

Cancel the common factor of $2 \cdot 0 - 0$ and $4 + 0$ .

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Step 7.1.1

Factor $- 1$ out of $2 \cdot 0$ .

$\frac{- (- 2 \cdot 0) - 0}{4 + 0}$

Step 7.1.2

Factor $- 1$ out of $- 0$ .

$\frac{- (- 2 \cdot 0) - (0)}{4 + 0}$

Step 7.1.3

Factor $- 1$ out of $- (- 2 \cdot 0) - (0)$ .

$\frac{- (- 2 \cdot 0 + 0)}{4 + 0}$

Step 7.1.4

Rewrite $- (- 2 \cdot 0 + 0)$ as $- 1 (- 2 \cdot 0 + 0)$ .

$\frac{- 1 (- 2 \cdot 0 + 0)}{4 + 0}$

Step 7.1.5

Factor $4$ out of $- 1 (- 2 \cdot 0 + 0)$ .

$\frac{4 (- 1 (- 2 \cdot 0 + 0))}{4 + 0}$

Step 7.1.6

Cancel the common factors.

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Step 7.1.6.1

Factor $4$ out of $4$ .

$\frac{4 (- 1 (- 2 \cdot 0 + 0))}{4 (1) + 0}$

Step 7.1.6.2

Factor $4$ out of $0$ .

$\frac{4 (- 1 (- 2 \cdot 0 + 0))}{4 \cdot 1 + 4 \cdot 0}$

Step 7.1.6.3

Factor $4$ out of $4 \cdot 1 + 4 \cdot 0$ .

$\frac{4 (- 1 (- 2 \cdot 0 + 0))}{4 \cdot (1 + 0)}$

Step 7.1.6.4

Cancel the common factor.

$\frac{4 (- 1 (- 2 \cdot 0 + 0))}{4 \cdot (1 + 0)}$

Step 7.1.6.5

Rewrite the expression.

$\frac{- 1 (- 2 \cdot 0 + 0)}{1 + 0}$

Step 7.2

Simplify the numerator.

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Step 7.2.1

Multiply $- 2$ by $0$ .

$\frac{- 1 (0 + 0)}{1 + 0}$

Step 7.2.2

Add $0$ and $0$ .

$\frac{- 1 \cdot 0}{1 + 0}$

Step 7.3

Add $1$ and $0$ .

$\frac{- 1 \cdot 0}{1}$

Step 7.4

Multiply $- 1$ by $0$ .

$\frac{0}{1}$

Step 7.5

Divide $0$ by $1$ .

$0$