Question
Find limits
√x²+100-10/x²
√x²+100-10/x²
Answers
GPT 3.5
To find the limits of the given expression, we need to evaluate the expression as x approaches both positive and negative infinity.
As x approaches positive infinity, we can simplify the expression as follows:
√(x² + 100) - (10 / x²)
= √(x² + 100) - 10 / x²
= (√(x² + 100) * x² - 10) / x²
= (x² + 100) * x² - 10 / x²
= x^4 + 100x² - 10 / x²
As x approaches positive infinity, both x² and x^4 grow infinitely large. Since the leading term in the numerator is x^4, the expression will also grow infinitely large. Therefore, the limit as x approaches positive infinity is positive infinity (∞).
As x approaches negative infinity, we can again simplify the expression as follows:
√(x² + 100) - 10 / x²
= (√(x² + 100) * x² - 10) / x²
= (x² + 100) * x² - 10 / x²
= x^4 + 100x² - 10 / x²
Again, as x approaches negative infinity, both x² and x^4 grow infinitely large. Since the leading term in the numerator is x^4, the expression will also grow infinitely large. However, as x approaches negative infinity, the expression becomes negative. Therefore, the limit as x approaches negative infinity is negative infinity (-∞).
In summary, we have:
lim(x→∞) (√(x² + 100) - 10 / x²) = ∞
lim(x→-∞) (√(x² + 100) - 10 / x²) = -∞
As x approaches positive infinity, we can simplify the expression as follows:
√(x² + 100) - (10 / x²)
= √(x² + 100) - 10 / x²
= (√(x² + 100) * x² - 10) / x²
= (x² + 100) * x² - 10 / x²
= x^4 + 100x² - 10 / x²
As x approaches positive infinity, both x² and x^4 grow infinitely large. Since the leading term in the numerator is x^4, the expression will also grow infinitely large. Therefore, the limit as x approaches positive infinity is positive infinity (∞).
As x approaches negative infinity, we can again simplify the expression as follows:
√(x² + 100) - 10 / x²
= (√(x² + 100) * x² - 10) / x²
= (x² + 100) * x² - 10 / x²
= x^4 + 100x² - 10 / x²
Again, as x approaches negative infinity, both x² and x^4 grow infinitely large. Since the leading term in the numerator is x^4, the expression will also grow infinitely large. However, as x approaches negative infinity, the expression becomes negative. Therefore, the limit as x approaches negative infinity is negative infinity (-∞).
In summary, we have:
lim(x→∞) (√(x² + 100) - 10 / x²) = ∞
lim(x→-∞) (√(x² + 100) - 10 / x²) = -∞