Find limits
√x²+100-10/x² where x approaches to 0
To find the limit, we substitute the value of x as it approaches 0 into the given expression:
lim(x→0) (√(x^2 +100) - 10/x^2)
When x approaches 0, we cannot directly substitute x = 0 into the expression, as it would result in division by zero. To overcome this, we can simplify the expression using algebra:
Let's manipulate the expression to help us get rid of the square root in the numerator:
lim(x→0) (√(x^2 +100) - 10/x^2)
lim(x→0) [(√(x^2 +100) - 10/x^2) * (√(x^2 +100) + 10/x^2)] / (√(x^2 +100) + 10/x^2)
Now, we can apply the difference of squares to simplify the expression further:
lim(x→0) [(x^2 +100 - (10/x^2)) / (√(x^2 +100) + 10/x^2)]
Next, we can multiply both the numerator and the denominator by the conjugate of the denominator, which is (√(x^2 +100) - 10/x^2):
lim(x→0) [(x^2 +100 - (10/x^2)) * (√(x^2 +100) - 10/x^2)] / [(√(x^2 +100) + 10/x^2) * (√(x^2 +100) - 10/x^2)]
Simplifying the expression:
lim(x→0) [(x^4 + 100x^2 - 10(x^2) - 100) / (x^2 + 10√(x^2 +100) + 100/x^2 - 100)]
Further simplification:
lim(x→0) [(x^4 - 10x^2 - 100) / (x^2 + 10√(x^2 +100) + 100/x^2 - 100)]
As x approaches 0, both x^2 and 1/x^2 approach infinity. Therefore, we have:
lim(x→0) [(∞ - ∞ - 100) / (∞ + 10√(∞ +100) + ∞ - 100)]
This expression is indeterminate, as we have (∞ - ∞) in the numerator and (∞ + ∞) in the denominator. To evaluate the limit further, we can apply L'Hôpital's Rule.
Differentiating the numerator and denominator with respect to x:
lim(x→0) [(8x^3 - 20x) / (2x - 20/√(x^2 +100) - 200/x^3)]
Now, substituting x = 0 into the new expression:
lim(x→0) [(0) / (2(0) - 20/√(0^2 +100) - 200/(0)^3)]
= 0/(-20 - 200/0)
= 0/(-∞)
= 0
Therefore, the limit of √(x^2 +100) - 10/x^2 as x approaches 0 is 0.
To find the limit of √(x² + 100) - (10/x²) as x approaches 0, we will need to simplify the expression.
First, let's simplify the term inside the square root:
√(x² + 100)
Since x is approaching 0, the value of x² will also approach 0. Therefore, we can simplify the expression as:
√(0 + 100) = √100 = 10
Next, let's simplify the other term:
10/x²
As x approaches 0, the denominator (x²) will approach 0. This means that the term will become undefined, as dividing by 0 is not possible. However, the numerator (10) is a constant, so we can consider it separately.
Now, let's put both simplified terms back into the original expression:
10 - (undefined)
Since we have an undefined term, the overall limit will also be undefined.
Therefore, the limit of √(x² + 100) - (10/x²) as x approaches 0 is undefined.