Two important limits, limx/sinx= 1 and limx/ln( 1+x)= 1, from which we can get x~ln( 1+x) and e x =1+x.

So the first question = lim (1-(1-x 2))/x 2 =1.

Second question = e lim [(LN2 * 2x+LN3 * 3x)/2] * [2/(2x+3x)]-L'H?pital's law.

=e^[(ln2+ln3)/2]

=e^ln√6

=√6

The third question =lim(tanx-x)/xtanxsinx

=limsec？ X/(tanxsinx+x (seconds? xsinx+tanxcosx))

= 1/0

=∞

The fourth question = lim (x+1+x) (2/x) = lim (1+2x) (1/2x * 4) = E4.