Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

$5 + 0 - 2 \cdot 0$

Step 8

Simplify the answer.

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Step 8.1

Multiply $- 2$ by $0$ .

$5 + 0 + 0$

Step 8.2

Add $5$ and $0$ .

$5 + 0$

Step 8.3

Add $5$ and $0$ .

$5$