Question

Evaluate `lim_(xrarr2)((e^(x)-e^(2))/(x-2)).`

Answer 1

Put `(x-2)=y,` so that when `x to2` then `y to0.`
`thereforelim_(xto2)((e^(x)-e^(2))/(x-2))=lim_(yto0)((e^(y+2)-e^(2))/(y))`
`=lim_(yto0){e^(2).((e^(y)-1)/(y))}`
`=e^(2).lim_(yto0)((e^(y)-1)/(y))=(e^(2)xx1)=e^(2)[becauselim_(yto0)((e^(y)-1)/(y))=1].`