Pulled this problem (which I missed) off my Calc "B" final from 40 years ago - answer is easy - problem is getting there in formal proof. Since I'm now semi-retired, thought I'd give it another shot.
Problem: Define ln(x) as the definite integral (1,x) of 1/t dt. Given that ln(x) is an increasing function and ln(1/2)<ln(x)<ln2, show (carefully) that ln(x) -> infinity as x -> infinity.
My solution (40 years late): Use the definition from Stewart, "Early Transcendentals," which says that you must show that for each positive number M such that f(x) > M, we have a corresponding positive number N with x>N.
Rough work: ln (x) > M. Since ln (x) is an increasing function (given), its inverse exists, which we know is exp(x). So ... taking exp of both sides of ln (x) > M, we get x > e**M. So choose N = e**M
Formal proof. ln (x) > M => x > e**M. So for each positive number M, choose a corresponding positive number N, such that N = e**M and we are done.
What bothers me: In my era, you always used all the information from the problem statement. Not sure of what the inequality ln(1/2)<ln(x)<ln 2, was supposed to suggest in terms of a problem solving strategy.
Anyway, how'd I do? Critique gratefully accepted.
gh02
Problem: Define ln(x) as the definite integral (1,x) of 1/t dt. Given that ln(x) is an increasing function and ln(1/2)<ln(x)<ln2, show (carefully) that ln(x) -> infinity as x -> infinity.
My solution (40 years late): Use the definition from Stewart, "Early Transcendentals," which says that you must show that for each positive number M such that f(x) > M, we have a corresponding positive number N with x>N.
Rough work: ln (x) > M. Since ln (x) is an increasing function (given), its inverse exists, which we know is exp(x). So ... taking exp of both sides of ln (x) > M, we get x > e**M. So choose N = e**M
Formal proof. ln (x) > M => x > e**M. So for each positive number M, choose a corresponding positive number N, such that N = e**M and we are done.
What bothers me: In my era, you always used all the information from the problem statement. Not sure of what the inequality ln(1/2)<ln(x)<ln 2, was supposed to suggest in terms of a problem solving strategy.
Anyway, how'd I do? Critique gratefully accepted.
gh02