Question

Find limit by using L'hopital's rule
lim ( 1 + lnx/x )
x→+♾️​

Answer 1

`\displaystyle \lim_(x\to \infty) 1+\cfrac{ln(x)}{x}\implies \lim_(x\to \infty) \cfrac{x+ln(x)}{x} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{ \textit{using L'Hopital once} }{\cfrac{(d)/(dx)[x+ln(x)]}{(d)/(dx)(x)}}\implies \cfrac{ ~~ 1+ ( 1 )/( x ) ~~ }{1}\implies 1+\cfrac{1}{x}\implies \cfrac{x+1}{x}`

`\stackrel{ \textit{using L'Hopital twice} }{\cfrac{(d)/(dx)[x+1]}{(d)/(dx)(x)}}\implies \cfrac{1}{1}\implies 1 \\\\[-0.35em] ~\dotfill\\\\ \displaystyle \lim_(x\to \infty) \cfrac{x+ln(x)}{x}\implies \lim_(x\to \infty) 1\implies \text{\LARGE 1}`