Question

Need to show work for limits

Answer 1

Answer:

$\displaystyle \lim_{x \to -\infty} (-7x^5 + x^3) = \infty$

General Formulas and Concepts:
Calculus

Limits

Limit Rule [Variable Direct Substitution]:
$\displaystyle \lim_{x \to c} x = c$

Step-by-step explanation:

Step 1: Define

Identify given.

$\displaystyle \lim_{x \to -\infty} (-7x^5 + x^3)$

Step 2: Evaluate

1. [Limit] Apply Limit Rule [Variable Direct Substitution]:
   $\displaystyle \lim_{x \to -\infty} (-7x^5 + x^3) = -7(- \infty)^5 + (- \infty)^3$

Recall that x⁵ is a "faster" function than x³. Also recall that we have 2 negatives, which would turn positive. Therefore, we can ignore the 2nd part of the limit and focus on the first:
$\displaystyle \begin{aligned}\lim_{x \to -\infty} (-7x^5 + x^3) & = -7(- \infty)^5 + (- \infty)^3 \\& = -7(- \infty)^5 \\& = -7(- \infty) \\& = 7(\infty) \\& = \boxed{\infty} \\\end{aligned}$

∴ we have found the limit to equal infinity.

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Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Limits