Calculus Examples

$lim_{x \to - \infty} 2 x^{- 8} + 4 x^{3}$

Step 1

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

$lim_{x \to - \infty} 2 x^{- 8} + lim_{x \to - \infty} 4 x^{3}$

Step 2

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

$2 lim_{x \to - \infty} x^{- 8} + lim_{x \to - \infty} 4 x^{3}$

Step 3

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

$2 lim_{x \to - \infty} \frac{1}{x^{8}} + lim_{x \to - \infty} 4 x^{3}$

Step 4

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{8}}$ approaches $0$ .

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

Step 5

Evaluate the limit.

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The limit at negative infinity of a polynomial of odd degree whose leading coefficient is positive is negative infinity.

$2 \cdot 0 + - \infty$

Simplify the answer.

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Multiply $2$ by $0$ .

$0 + - \infty$

Subtract $\infty$ from $0$ .

$- \infty$