Evaluate the Limit

Problem

(lim_t→0)(5/(t√(,1+t))−5/t)

Solution

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

5/(t√(,1+t))−(5√(,1+t))/(t√(,1+t))

2. Combine the numerators over the shared denominator.

(5−5√(,1+t))/(t√(,1+t))

3. Factor out the constant 5 from the numerator to simplify the expression.

(5*(1−√(,1+t)))/(t√(,1+t))

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

(5*(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

(5*(1−(1+t)))/(t√(,1+t)*(1+√(,1+t)))

6. Distribute and simplify the terms inside the parentheses in the numerator.

(5*(−t))/(t√(,1+t)*(1+√(,1+t)))

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

(−5)/(√(,1+t)*(1+√(,1+t)))

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

(−5)/(√(,1+0)*(1+√(,1+0)))

9. Calculate the final value by simplifying the arithmetic.

(−5)/(1*(1+1))=−5/2

Final Answer

(lim_t→0)(5/(t√(,1+t))−5/t)=−5/2

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