Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Evaluate the limit of $5$ which is constant as $x$ approaches $\infty$ .

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

Step 4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 5

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

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

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 $\infty$ .

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

Step 6.2

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

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

Step 7

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 8

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

$\frac{5 + 0}{3 \cdot 0 + lim_{x \to \infty} 1}$

Step 9

Evaluate the limit.

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Step 9.1

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

$\frac{5 + 0}{3 \cdot 0 + 1}$

Step 9.2

Simplify the answer.

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Step 9.2.1

Add $5$ and $0$ .

$\frac{5}{3 \cdot 0 + 1}$

Step 9.2.2

Simplify the denominator.

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Step 9.2.2.1

Multiply $3$ by $0$ .

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

Step 9.2.2.2

Add $0$ and $1$ .

$\frac{5}{1}$

Step 9.2.3

Divide $5$ by $1$ .

$5$