Question

How do you find `\lim _ { x \rightarrow 0} \frac { x + \sin ( x ) } { x }`?

Answer 1

2

Explanation:

You would substitute the expression:

`(x+sinx)/x=x/x+sinx/x=1+sinx/x`

Then you will find that

`lim_(x->0)(1+sinx/x)=1+1=2`

Answer 2

The limit is `2`.

Explanation:

`lim_(x->0)(x+sinx)/x=0/0`

Since it is in the indeterminate form `0/0` you may apply l'Hospital's Rule, that is given `f(a)` and `f(b)` is differentiable:

`(lim_(x->c)f'(a))/(lim_(x->c)f'(b))=(lim_(x->0)d/dx[x+sinx])/(lim_(x->c)d/dx[x])=(lim_(x->0)(1+cosx))/(lim_(x->c)1)=(1+cos(0))/1=(1+1)/1=2`

Thus the limit is `2`.