Evaluate the Limit

Problem

(lim_t→∞)(√(,t+t2)/(7*t−t2))

Solution

1. Identify the highest power of t in the denominator to simplify the limit at infinity. The highest power is t2

2. Divide both the numerator and the denominator by t2 to normalize the expression.

(lim_t→∞)(√(,t+t2)/(t2)/(7*t−t2)/(t2))

3. Simplify the denominator by dividing each term by t2

(7*t)/(t2)−(t2)/(t2)=7/t−1

4. Rewrite the numerator by bringing t2 inside the square root as t4

√(,t+t2)/(t2)=√(,(t+t2)/(t4))

5. Simplify the expression inside the square root.

√(,t/(t4)+(t2)/(t4))=√(,1/(t3)+1/(t2))

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

(lim_t→∞)(√(,1/(t3)+1/(t2))/(7/t−1))=√(,0+0)/(0−1)

7. Calculate the final value.

0/(−1)=0

Final Answer

(lim_t→∞)(√(,t+t2)/(7*t−t2))=0

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