Evaluate the Limit

Problem

(lim_t→∞)(√(,t+t^2)/(2*t−t^2))

Solution

1. Identify the highest power of t in the denominator to simplify the limit as t approaches infinity. The highest power is t^2

2. Divide both the numerator and the denominator by t^2 to normalize the expression.

√(,t+t^2)/(t^2)/(2*t−t^2)/(t^2)

3. Simplify the numerator by bringing t^2 inside the square root as t^4

√(,t/(t^4)+(t^2)/(t^4))/((2*t)/(t^2)−(t^2)/(t^2))

4. Reduce the fractions within the expression.

√(,1/(t^3)+1/(t^2))/(2/t−1)

5. Evaluate the limit by observing that as t→∞ terms with t in the denominator approach 0

√(,0+0)/(0−1)

6. Calculate the final numerical value.

0/(−1)=0

Final Answer

(lim_t→∞)(√(,t+t^2)/(2*t−t^2))=0

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