Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

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 $\infty$ .

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

Step 6.2

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

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

Step 6.3

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

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify the answer.

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Step 9.1

Add $5$ and $0$ .

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

Step 9.2

Simplify the denominator.

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Step 9.2.1

Multiply $3$ by $0$ .

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

Step 9.2.2

Add $1$ and $0$ .

$\frac{5}{1}$

Step 9.3

Divide $5$ by $1$ .

$5$