Find limits
Square root x²+100-10/x² where x approaches to 0
To find the limit of the expression (√(x² + 100) - 10) / x² as x approaches 0, we can simplify and then substitute x = 0 into the expression.
First, let's simplify the expression (√(x² + 100) - 10) / x² using some algebraic manipulations:
= (√(x² + 100) - 10) / x²
= ((√(x² + 100) - 10) / x²) * (√(x² + 100) + 10) / (√(x² + 100) + 10)
= ((x² + 100 - 100) / (x^2 * (√(x² + 100) + 10))
= (x² / (x^2 * (√(x² + 100) + 10))
= 1 / (√(x² + 100) + 10)
Now we can substitute x = 0 into the expression:
lim(x→0) 1 / (√(x² + 100) + 10)
= 1 / (√(0² + 100) + 10)
= 1 / (√100 + 10)
= 1 / (10 + 10)
= 1 / 20
= 0.05
Therefore, the limit of (√(x² + 100) - 10) / x² as x approaches 0 is 0.05.
To find the limit of the expression √(x^2 + 100) - 10/x^2 as x approaches 0, we can simplify the expression and then substitute the value of x into the simplified form.
First, let's simplify the expression √(x^2 + 100):
√(x^2 + 100) = √(x^2) * √(1 + 100/x^2)
Since √(x^2) is equal to |x| when x is a real number, we can rewrite the expression as:
|x| * √(1 + 100/x^2)
Now, let's substitute x = 0 into the simplified expression:
|x| * √(1 + 100/0^2) = |0| * √(1 + 100/0) = 0 * √(1 + ∞) = 0 * ∞
Here, we have an indeterminate form, as the product of 0 and infinity is not well-defined.
To determine the limit, we need to investigate the behavior of the expression as x approaches 0 from both the positive and negative sides.
When x approaches 0 from the positive side (x > 0), √(x^2 + 100) - 10/x^2 approaches ∞.
When x approaches 0 from the negative side (x < 0), √(x^2 + 100) - 10/x^2 approaches -∞.
Therefore, the limit of the expression √(x^2 + 100) - 10/x^2 as x approaches 0 does not exist.