Question

Evaluate the following limit. Enter the exact answer. To enter ✓a, type sqrt(a). lim x-0 √x+5-√5 27 Show your work and explain.

Answer 1

The limit as x approaches 0 for √(x+5) - √5 is calculated by multiplying and dividing by the conjugate, simplifying to x/x, which approaches 1 as x approaches 0.

Step-by-step explanation:

The question involves finding the limit as x approaches 0 for the expression √(x+5) - √5. To evaluate this limit, a common approach is to rationalize the numerator by multiplying and dividing by the conjugate of the numerator, which in this case would be √(x+5) + √5. The purpose of this method is to remove the radical in the numerator and simplify the expression. After the multiplication, the terms involving the square roots in the numerator will cancel out, leaving a difference of squares which simplifies to x in the numerator and a linear expression involving √(x+5) + √5 in the denominator. As we let x approach 0, the expression in the denominator will approach 2√5. Therefore, the limit can now be easily evaluated.

To calculate the limit step by step:

1. Multiply the original expression by (√(x+5) + √5)/(√(x+5) + √5).
2. Simplify the numerator to obtain x.
3. The denominator will be (x+5) - 5 after simplifying, which simplifies further to x.
4. As x approaches 0, the limit of x/x is 1.