Find the limits
√x²+100 -10 /x²
where x approaches to 0
To find the limit of the given expression as x approaches 0, we can simplify it and then substitute 0 for x.
First, let's simplify the expression:
√(x²+100) - 10 / x²
Multiply the numerator and denominator by the conjugate of the square root:
[√(x²+100) - 10] / x² ⋅ [(√(x²+100) + 10) / (√(x²+100) + 10)]
Use the difference of squares:
[(x² + 100) - 10√(x²+100)] / x² ⋅ [(√(x²+100) + 10) / (√(x²+100) + 10)]
Multiply the numerators and denominators together:
[(x² + 100) - 10√(x²+100)][(√(x²+100) + 10)] / [x²(√(x²+100) + 10)]
Now, let's substitute 0 for x:
[(0² + 100) - 10√(0²+100)][(√(0²+100) + 10)] / [0²(√(0²+100) + 10)]
Simplifying further, we get:
[(100 - 10√100)][(10)] / [0]
Since √(100) = 10,
[100 - 10(10)][10] / [0]
Simplifying the numerator and denominator:
[100 - 100][10] / [0]
This gives us:
[0][10] / [0]
Since we have 0 in the numerator and denominator, the expression becomes undefined.
Therefore, the limit of √x²+100 -10 / x² as x approaches 0 is undefined.
To find the limit of the expression ( √x² + 100 - 10 ) / x² as x approaches 0, follow these steps:
Step 1: Simplify the expression
The expression can be simplified by using the rules of square roots. Since x² is always non-negative, we can simplify √(x² + 100) as follows:
√(x² + 100) = √(x²) * √(1 + 100/x²) = x * √(1 + 100/x²)
Now the expression becomes:
[ x * √(1 + 100/x²) - 10 ] / x²
Step 2: Cancel out the common factor
Since both the numerator and denominator contain a factor of x, we can cancel them out:
[ √(1 + 100/x²) - 10/x ] / x
Step 3: Evaluate the limit
Now that the expression has been simplified, we can evaluate the limit as x approaches 0. Let's substitute x = 0 into the simplified expression:
[ √(1 + 100/0²) - 10/0 ] / 0
We notice that both the numerator and denominator have an undefined value when x = 0. In such cases, we need to apply other techniques to determine the limit, such as L'Hôpital's Rule or factoring techniques.
Therefore, the limit of the expression as x approaches 0 is undefined.