Evaluate the Limit

Problem

(lim_x→0)(1/(x√(,1+x))−1/x)

Solution

1. Find a common denominator to combine the two fractions into a single expression.

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

2. Rationalize the numerator by multiplying the numerator and the denominator by the conjugate of the numerator, which is 1+√(,1+x)

(1−√(,1+x))/(x√(,1+x))⋅(1+√(,1+x))/(1+√(,1+x))=(1−(√(,1+x))2)/(x√(,1+x)*(1+√(,1+x)))

3. Simplify the numerator by expanding the squares and combining like terms.

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

4. Cancel the common factor of x from the numerator and the denominator, provided x≠0

(−1)/(√(,1+x)*(1+√(,1+x)))

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

(−1)/(√(,1+0)*(1+√(,1+0)))=(−1)/(1*(1+1))

6. Simplify the result to find the final numerical value.

(−1)/2

Final Answer

(lim_x→0)(1/(x√(,1+x))−1/x)=−1/2

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