lim(x→a)+ cos(x) ln(x−a)/ln(e^x−e^a)

Answer 1

limcosx(ln(x - a)/ln(e^x - e^a)) as x → a.

Substitution give us indeterminatate form ∞/∞.

We can apply Lopital rule: limcosx·(ln(x - a)/ln(e^x - e^a)) = cosa·lim((ln(x - a))'/(ln(e^x - e^a)) = cosa·lim(1/(x - a):e^x/(e^x - e^a)) = cosa·lim((e^x - e^a):(x - a)/e^x) = cosa·e^a/e^a = cosa, all limits as x → a;

Answer: this limit equel cosa.