Find all values of a so that lim as x approaches infinity of ((x+a)/(x-a))^x= e

Question

Explanation / Answer

example Find all values of a so that lim x-> infinity [(x+a)/(x-a)]^x = e? lim x-> infinity [(x+a)/(x-a)]^x = e solutioin Let L = lim(x?8) [(x+a)/(x-a)]^x. Take ln's of both sides: ln L = lim(x?8) x ln[(x+a)/(x-a)] ......= lim(x?8) ln[(x+a)/(x-a)] / x^(-1), which is of the form 0/0 Since (d/dx) ln[(x+a)/(x-a)] = (d/dx) [ln(x+a) - ln(x-a)] = 1/(x+a) - 1/(x-a) = -2a/(x^2 - a^2), applying L'Hopital's Rule yields ln L = lim(x?8) [-2a/(x^2 - a^2)] / (-x^(-2)) ......= lim(x?8) 2ax^2 / (x^2 - a^2) ......= lim(x?8) 2a / (1 - a^2/x^2) ......= 2a. Hence, L = e^(2a). Since we want the limit to equal e, we conclude that 2a = 1 ==> a = 1/2. hope iit helps

Related Questions

Navigate

• Browse (All)
• Browse F
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.