Find limits
√x²+100-10/x²
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) = -∞
To find the limit of the expression (√x² + 100 - 10) / x² as x approaches a certain value, we can simplify the expression first.
Let's start by simplifying the numerator (√x² + 100 - 10):
√x² + 100 - 10 = √x² + 90
Now we have the expression (√x² + 90) / x².
To find the limit as x approaches a value, let's substitute that value into the expression:
lim (x→a) (√x² + 90) / x²
If a = 0, substituting it directly into the expression would result in division by zero, which is undefined. In this case, we need to use a different method.
We can rewrite the expression as follows:
lim (x→0) (√x² + 90) / x² = lim (x→0) (√x²/x² + 90/x²)
Now we can simplify further:
lim (x→0) (√1 + 90/x²)
As x approaches 0, 90/x² becomes infinitely large, and the limit of this expression is positive infinity (∞).
Therefore, the limit of (√x² + 100 - 10) / x² as x approaches 0 is positive infinity (∞).