Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

$3 \cdot 0 + 4 \cdot 0 + lim_{x \to \infty} 5$

Step 8

Evaluate the limit.

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Step 8.1

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

$3 \cdot 0 + 4 \cdot 0 + 5$

Step 8.2

Simplify the answer.

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

Multiply $3$ by $0$ .

$0 + 4 \cdot 0 + 5$

Step 8.2.1.2

Multiply $4$ by $0$ .

$0 + 0 + 5$

Step 8.2.2

Add $0$ and $0$ .

$0 + 5$

Step 8.2.3

Add $0$ and $5$ .

$5$