Lim (√(3-x) - √(3+x)) /x
x->0
-1/sqrt(3)
I think you wanted to know how:
multily numerator and denom by
sqrt(3-x) + sqrt(3+x)
lim ( 3-x -3-x) )/(x(sqrt(3-x) + sqrt(3+x))
lim(-2x/x(sqrt(3-x) + sqrt(3+x)))
lim -2/(sqrt(3-x) + sqrt(3+x))
=-1/sqrt(3)
To evaluate the limit of the given expression as x approaches 0, we can apply algebraic manipulations and the limit properties. Here's a step-by-step process:
Step 1: Simplify the expression.
We have (√(3-x) - √(3+x)) / x. To simplify, we can rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is (√(3-x) + √(3+x)). This allows us to eliminate the square roots in the numerator.
√(3-x) - √(3+x) = (√(3-x) - √(3+x)) * (√(3-x) + √(3+x))
Step 2: Apply the difference of squares.
Using the difference of squares formula, we can simplify (√(3-x) - √(3+x)) * (√(3-x) + √(3+x)) to (3-x) - (3+x).
(3 - x) - (3 + x) = 3 - x - 3 - x = -2x
Step 3: Rewrite the expression.
The expression (√(3-x) - √(3+x)) / x can now be rewritten as -2x / x.
Step 4: Cancel out the "x" terms.
Since we have -2x in the numerator and x in the denominator, we can cancel out an "x," leaving -2.
Step 5: Evaluate the limit.
Now that we have simplified the expression, we can evaluate the limit:
lim (x->0) (-2) = -2
Therefore, the limit of the expression (√(3-x) - √(3+x)) / x as x approaches 0 is -2.