हम जानते है कि
`e^(x)=1+x+(x^(2))/(2!)+(x^(3))/(3!)+(x^(4))/(4!)+...oo`
तथा `e^(-x)=1-x+(x^(2))/(2!)-(x^(3))/(3!)+(x^(4))/(4!)...oo`
`therefore e^(x)-e^(-x)=2(x+(x^(3))/(3!)+(x^(5))/(5!)+...oo)`
`=2x(1+(x^(2))/(3!)+(x^(4))/(5!)+...oo)`
`therefore (e^(x)-e^(-x))/(x)=2[1+(x^(2))/(3!)+(x^(4))/(5!)+...oo]`
अतः `underset(xto0)lim(e^(x)-e^(-x))/(x)=underset(xto0)lim2[1+(x^(2))/(3!)+(x^(4))/(5!)+...oo]`
`=2[1+0+0+...]` (सिमा लेने पर )
`=2`
