Evaluate the Limit

Problem

(lim_x→0)((√(,1+x)−√(,1−x))/x)

Solution

1. Identify the indeterminate form by substituting x=0 into the expression, which results in 0/0

2. Rationalize the numerator by multiplying both the numerator and the denominator by the conjugate expression √(,1+x)+√(,1−x)

3. Expand the numerator using the difference of squares formula (a−b)*(a+b)=a2−b2

((√(,1+x)−√(,1−x))*(√(,1+x)+√(,1−x)))/(x*(√(,1+x)+√(,1−x)))

4. Simplify the numerator by removing the square roots and combining like terms.

((1+x)−(1−x))/(x*(√(,1+x)+√(,1−x)))

(2*x)/(x*(√(,1+x)+√(,1−x)))

5. Cancel the common factor of x from the numerator and the denominator.

2/(√(,1+x)+√(,1−x))

6. Evaluate the limit by substituting x=0 into the simplified expression.

2/(√(,1+0)+√(,1−0))

2/(1+1)

Final Answer

(lim_x→0)((√(,1+x)−√(,1−x))/x)=1

Want more problems? Check here!