Question

How do you find `lim_(x->∞)(x+2sinx-1)/(x+3cosx+1)`?

How do you find `lim_(x->∞)(x+2sinx-1)/(x+3cosx+1)`?

Answer 1

`lim_(xrarr+oo)(x+2sinx-1)/(x+3cosx+1)=1`

Explanation:

`lim_(xrarr+oo)(x+2sinx-1)/(x+3cosx+1)`

`x->+oo`
`x>0`

`=lim_(xrarr+oo)(1+2sinx/x-1/x)/(1+3cosx/x+1/x)=1`

because

• `lim_(xrarr+oo)1/x=^((1/(+oo)))0`

• `lim_(xrarr+oo)sinx/x=0`

`|sinx/x|<=1/|x|`
so

`-1/|x|<=sinx/x<=1/|x|`

Using the sandwich/squeeze theorem we get:

`lim_(xrarr+oo)-1/|x|=0=lim_(xrarr+oo)1/|x|`

Therefore ,

`lim_(xrarr+oo)sinx/x=0`

• `lim_(xrarr+oo)cosx/x=0`

`|cosx/x|<=1/|x|`

`-1/|x|<=cosx/x<=1/|x|`

`lim_(xrarr+oo)-1/|x|=0=lim_(xrarr+oo)1/|x|`

Therefore,

`lim_(xrarr+oo)cosx/x=0`