Evaluate the Limit

Problem

(lim_t→0)(7/(t√(,1+t))−7/t)

Solution

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

7/(t√(,1+t))−(7√(,1+t))/(t√(,1+t))

2. Combine the numerators over the shared denominator.

(7−7√(,1+t))/(t√(,1+t))

3. Factor out the constant 7 from the numerator to simplify the expression.

(7*(1−√(,1+t)))/(t√(,1+t))

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

(7*(1−√(,1+t))*(1+√(,1+t)))/(t√(,1+t)*(1+√(,1+t)))

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

(7*(1−(1+t)))/(t√(,1+t)*(1+√(,1+t)))

6. Distribute and subtract in the numerator.

(7*(−t))/(t√(,1+t)*(1+√(,1+t)))

7. Cancel the common factor t from the numerator and the denominator, noting that t≠0 as it approaches the limit.

(−7)/(√(,1+t)*(1+√(,1+t)))

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

(−7)/(√(,1+0)*(1+√(,1+0)))

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

(−7)/(1*(1+1))=−7/2

Final Answer

(lim_t→0)(7/(t√(,1+t))−7/t)=−7/2

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