Evaluate the Limit

Problem

(lim_t→0)(1/(t√(,1+t))−1/t)

Solution

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

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

2. Rationalize the numerator by multiplying the numerator and the denominator by the conjugate, 1+√(,1+t)

(1−√(,1+t))/(t√(,1+t))⋅(1+√(,1+t))/(1+√(,1+t))=(1−(1+t))/(t√(,1+t)*(1+√(,1+t)))

3. Simplify the numerator by performing the subtraction.

(−t)/(t√(,1+t)*(1+√(,1+t)))

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

(−1)/(√(,1+t)*(1+√(,1+t)))

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

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

6. Simplify the result to find the final value.

(−1)/2

Final Answer

(lim_t→0)(1/(t√(,1+t))−1/t)=−1/2

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