How do you find the limit of ( x + sin(x) ) / (x) as x approaches 0?

2 Answers
May 22, 2018

2.

Explanation:

Knowing that, lim_(x to 0)sinx/x=1, we have,

lim_(x to 0)(x+sinx)/x,

=lim_(x to 0){x/x+sinx/x},

=lim_(x to 0){1+sinx/x},

=1+1

=2.

May 22, 2018

color(blue)[lim_(xrarr0)x/x+lim_(xrarr0)sinx/x=1+1=2]

Explanation:

Note that:

color(red)[lim_(nrarr0)sinn/n=1

lim_(xrarr0)( x + sin(x) ) / (x)

lim_(xrarr0)x/x+lim_(xrarr0)sinx/x

lim_(xrarr0)x/x=0/0

since the direct compensation product equal 0/0
we will use L'hospital Rule.

L'hospital Rule color(red)[lim_(trarra)(f'(x))/(g('x))]

lim_(xrarr0)x/x=lim_(xrarr0)1/1=1

lim_(xrarr0)sinx/x=1

lim_(xrarr0)sinx/x=1

color(blue)[lim_(xrarr0)x/x+lim_(xrarr0)sinx/x=1+1=2]