Evaluate the limit. Verify your answer by using a table of values or a graph.
lim [radical(3+x)-radical(3)] / x
x->0
To evaluate the limit of the given expression, we can apply some algebraic manipulations and then use the properties of limits. Let's break it down step by step:
1. Begin by simplifying the expression:
lim as x approaches 0 [√(3 + x) - √3] / x
2. We want to eliminate the square roots from the numerator. Multiply the expression by the conjugate of the numerator to get rid of the radicals:
lim as x approaches 0 [(√(3 + x) - √3) * (√(3 + x) + √3)] / (x * (√(3 + x) + √3))
3. Simplify the numerator after multiplying using the difference of squares:
lim as x approaches 0 [(√(3 + x))^2 - (√3)^2] / (x * (√(3 + x) + √3))
= lim as x approaches 0 (3 + x - 3) / (x * (√(3 + x) + √3))
= lim as x approaches 0 x / (x * (√(3 + x) + √3))
4. Cancel out the x terms in the denominator:
lim as x approaches 0 1 / (√(3 + x) + √3)
5. Now, let's evaluate the limit by substituting the value 0 into the expression:
1 / (√(3 + 0) + √3)
= 1 / (√3 + √3)
= 1 / (2√3)
So, the limit of the given expression as x approaches 0 is 1 / (2√3).
To verify this answer, you can create a table of values or use a graphing calculator to evaluate the expression for various values of x approaching 0.