Making use of the method of replacing an infinitesimal with an equivalent one, find the following limits:
`underset(x to 0)lim (sin 3x)/(ln (1+5x))` (b) `underset(x to 0)lim (ln (1+sin 4x))/(e^(sin 5x)-1)`
(c ) `underset(x to 0) (e^(sin 3x)-1)/(ln (1+tan 2x))` (d) `underset(x to 0)lim (arc tan 3x)/(arc sin 2x)`
(e ) `underset(x to 0)lim (ln (2-cos 2x))/(ln^(2) (sin 3x+1))` (f) `underset(x to 0)lim (sqrt(1+sin 3x)-1)/(ln (1+tan 2x))`
(g) `underset(x to 0)lim ln (1+2x-3x^(2)+4x^(3))/(ln (1-x+2x^(2)-7x^(3))` (h) `underset(x to 0)lim (sqrt(1+x^(2))-1)/(1-cos x)`

Answer

`(a) 3/5, (b) 4/5, (c) 3/2, (d) 3/2, (e) 2/9, (f) 3/4, (g)-2, (h)1`.