Limit of a Polynomial at Infinity Worksheet (with solutions ... - Free Printable

You're working on finding the limit of a polynomial at infinity — specifically, as $ x \to +\infty $ or $ x \to -\infty $. The key idea is that for polynomials, the term with the highest degree dominates the behavior as $ x $ becomes very large (positive or negative). So we focus on the leading term.

Let’s solve each part step by step:

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i) $ \lim_{x \to +\infty} (-2x^3 + 5x^2 + 4x - 7) $

- Leading term: $ -2x^3 $
- Degree = 3 (odd), coefficient = -2 (negative)
- As $ x \to +\infty $, $ x^3 \to +\infty $, so $ -2x^3 \to -\infty $
- Lower-degree terms become negligible compared to $ -2x^3 $

✔ Answer: $ \boxed{-\infty} $

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j) $ \lim_{x \to -\infty} (x^4 - 5x) $

- Leading term: $ x^4 $
- Degree = 4 (even), coefficient = +1 (positive)
- As $ x \to -\infty $, $ x^4 \to +\infty $ (since even power makes it positive)
- $ -5x $ becomes $ +\infty $ too (because $ x $ is negative, so $ -5x > 0 $), but it’s dominated by $ x^4 $

✔ Answer: $ \boxed{+\infty} $

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k) $ \lim_{x \to +\infty} (4x - 2x^4) $

- Leading term: $ -2x^4 $
- Degree = 4 (even), coefficient = -2 (negative)
- As $ x \to +\infty $, $ x^4 \to +\infty $, so $ -2x^4 \to -\infty $
- The $ 4x $ term is negligible

✔ Answer: $ \boxed{-\infty} $

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## ✔ Final Answers:

- i) $ \boxed{-\infty} $
- j) $ \boxed{+\infty} $
- k) $ \boxed{-\infty} $

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## 💡 Quick Rule Summary:

For a polynomial $ P(x) = a_n x^n + \dots + a_0 $:

- As $ x \to \pm\infty $, $ P(x) \approx a_n x^n $
- If $ n $ is even:
- $ a_n > 0 $ → limit is $ +\infty $ in both directions
- $ a_n < 0 $ → limit is $ -\infty $ in both directions
- If $ n $ is odd:
- $ a_n > 0 $ → $ +\infty $ as $ x \to +\infty $, $ -\infty $ as $ x \to -\infty $
- $ a_n < 0 $ → $ -\infty $ as $ x \to +\infty $, $ +\infty $ as $ x \to -\infty $

This rule lets you solve these quickly without expanding or calculating!

Let me know if you’d like to see graphical intuition or more examples!