Calculus Examples

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

Step 1

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

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

Step 2

Evaluate the limit.

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Step 2.1

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

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Step 2.1.1

Raise $x$ to the power of $1$ .

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

Step 2.1.2

Factor $x$ out of $x^{1}$ .

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

Step 2.1.3

Cancel the common factors.

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Step 2.1.3.1

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

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

Step 2.1.3.2

Cancel the common factor.

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

Step 2.1.3.3

Rewrite the expression.

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

Step 2.2

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

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Divide $2$ by $1$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Evaluate the limit.

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Step 6.1

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

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

Step 6.2

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

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

Step 7

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

$\frac{0 + 5 \cdot 0}{2 + 0}$

Step 8

Simplify the answer.

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Step 8.1

Cancel the common factor of $0 + 5 \cdot 0$ and $2 + 0$ .

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Step 8.1.1

Reorder terms.

$\frac{0 + 0 \cdot 5}{2 + 0}$

Step 8.1.2

Factor $2$ out of $0$ .

$\frac{2 (0) + 0 \cdot 5}{2 + 0}$

Step 8.1.3

Factor $2$ out of $0 \cdot 5$ .

$\frac{2 (0) + 2 (0 \cdot 5)}{2 + 0}$

Step 8.1.4

Factor $2$ out of $2 (0) + 2 (0 \cdot 5)$ .

$\frac{2 (0 + 0 \cdot 5)}{2 + 0}$

Step 8.1.5

Cancel the common factors.

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Step 8.1.5.1

Factor $2$ out of $2$ .

$\frac{2 (0 + 0 \cdot 5)}{2 \cdot 1 + 0}$

Step 8.1.5.2

Factor $2$ out of $0$ .

$\frac{2 (0 + 0 \cdot 5)}{2 \cdot 1 + 2 \cdot 0}$

Step 8.1.5.3

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

$\frac{2 (0 + 0 \cdot 5)}{2 \cdot (1 + 0)}$

Step 8.1.5.4

Cancel the common factor.

$\frac{2 (0 + 0 \cdot 5)}{2 \cdot (1 + 0)}$

Step 8.1.5.5

Rewrite the expression.

$\frac{0 + 0 \cdot 5}{1 + 0}$

Step 8.2

Simplify the numerator.

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Step 8.2.1

Multiply $0$ by $5$ .

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

Step 8.2.2

Add $0$ and $0$ .

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

Step 8.3

Add $1$ and $0$ .

$\frac{0}{1}$

Step 8.4

Divide $0$ by $1$ .

$0$