Question

What are the limits?

A)`lim_(x->0)(e^x+x-1)/(1-e^-x`
B)`lim_(x->1)(1-x^3)/(2-sqrt(x^2+3)`
C)`lim_(x->0)(sin(2x)+tanx)/(6x)`

Answer 1

The answer
a)`lim_(x->0)(e^x+x-1)/(1-e^-x)=2`

b)b)`lim_(x->1)(1-x^3)/(2-sqrt(x^2+3))=6`

c)`lim_(x->0)(sin(2x)+tanx)/(6x)=1/2`

Explanation:

Direct compensation product equal `0/0` so we should applied laplace law

if Direct compensation product equal `0/0` laplace law is

`lim_(xrarra)[(f(x)')/(g(x)')]`

a)`lim_(x->0)(e^x+x-1)/(1-e^-x)=lim_(x->0)(e^x+1)/(e^-x)=lim_(x->0)(e^0+1)/(e^-0)=(1+1)/1=2`

b)`lim_(x->1)(1-x^3)/(2-sqrt(x^2+3))=lim_(x->1)(-3x^2)/((-2x)/(2sqrt(x^2+3)))=sqrt(9x^4+27x^2)=sqrt36=6`

c)`lim_(x->0)(sin(2x)+tanx)/(6x)=lim_(x->0)[(2cos(2x))+sec^2(x)]/6=(2+1)/6=3/6=1/2`