Question

I need some help solving two limits:

Answer 1

1) \[\lim_{x \rightarrow +\infty} \left[ \ln (x ^{2}+x)-\ln x \right]x\]
2)
\[\lim_{x \rightarrow +\infty} (x \sqrt{x ^{2}-1}-\sqrt{x ^{4}+x ^{3}})\]

Any help, advice would be refreshing.

Answer 2

for the 2nd one,
put x = 1/y
as x->infinity , y-> 0
simplify after you put x=1/y, x^2=1/y^2 ,...

Answer 3

for 1), combine the log expression into ln((x^2+x)/x), or ln(x+1). now, ln(x+1)*x will approach infinity as x->infinity since log(x+1) and x will both approach infinity.