Find the limits
underroot x²+100 -10/x²
where x approaches to 0
To find the limit as x approaches 0 of the expression √(x²+100) - (10/x²), we can simplify it as follows:
We first calculate the limit of each term separately.
1. Limit as x approaches 0 of √(x² + 100):
√(0² + 100) = √100 = 10.
2. Limit as x approaches 0 of 10/x²:
10/0² is undefined.
Therefore, the limit of the expression as x approaches 0 does not exist.
To find the limit of the given expression as x approaches 0, we can start by simplifying the expression.
Taking the square root of x^2 + 100 can be simplified as |x|√(1 + 100/x^2), where |x| represents the absolute value of x.
The expression becomes:
|x|√(1 + 100/x^2) - 10/x^2
Next, we can simplify further by splitting the limit into two parts, since the limit of a sum is the sum of the limits:
lim(x→0) [|x|√(1 + 100/x^2)] - lim(x→0) [10/x^2]
Now, let's evaluate each limit separately:
First, let's consider the limit as x approaches 0 of [|x|√(1 + 100/x^2)].
Since the absolute value of x approaches 0 from both positive and negative sides, and 1 + 100/x^2 approaches 1 as x approaches 0, we can rewrite the expression as:
lim(x→0) |x| * √(1 + 100/x^2)
Now, an important property to note is that the limit of a product is the product of the limits. Therefore, we can rewrite the expression as:
lim(x→0) |x| * lim(x→0) √(1 + 100/x^2)
As x approaches 0, |x| approaches 0 as well. So, we have:
0 * lim(x→0) √(1 + 100/x^2)
0 multiplied by any number is still 0, so the limit of the first part is 0.
Now, let's evaluate the second part of the expression:
lim(x→0) [10/x^2]
As x approaches 0, the denominator (x^2) approaches 0, and the numerator (10) remains constant. So, we have:
10/0^2
Any number divided by zero is undefined, so this limit does not exist.
Since the first part of the expression evaluates to 0 and the second part is undefined, the overall limit of the expression as x approaches 0 is undefined.