Evaluate the Limit

Problem

(lim_t→∞)(√(,t+t^2)/(7*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)/(7*t−t^2)/(t^2))

3. Simplify the denominator by dividing each term by t^2

(7*t)/(t^2)−(t^2)/(t^2)=7/t−1

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

√(,t+t^2)/(t^2)=√(,(t+t^2)/(t^4))

5. Simplify the expression inside the square root.

√(,t/(t^4)+(t^2)/(t^4))=√(,1/(t^3)+1/(t^2))

6. Apply the limit as t→∞ Recall that (lim_t→∞)(1/(t^n))=0 for n>0

(lim_t→∞)(√(,1/(t^3)+1/(t^2))/(7/t−1))=√(,0+0)/(0−1)

7. Calculate the final value.

0/(−1)=0

Final Answer

(lim_t→∞)(√(,t+t^2)/(7*t−t^2))=0

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