Evaluate the Limit

Problem

(lim_t→∞)(√(,t+t^2)/(4*t−t^2))

Solution

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

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

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

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

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

4. Reduce the fractions inside the expression.

(lim_t→∞)(√(,1/(t^3)+1/(t^2))/(4/t−1))

5. Evaluate the limit by applying the property that (lim_t→∞)(1/(t^n))=0 for n>0

√(,0+0)/(0−1)

6. Calculate the final numerical value.

0/(−1)=0

Final Answer

(lim_t→∞)(√(,t+t^2)/(4*t−t^2))=0

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