Evaluate: `lim_(x->oo)` `(x+7sinx)/(-2x+13)`

Explanation

We known that `-1lesinxle1`for all x. So,
`-7le7sinxlex+7`
or `x-7lex+7sinxlex+7`
Dividing throughout by `-2x+13,` we get
`(x-7)/(-2x+13)ge(x+7sinx)/(-2x+13)ge(x+7)/(-2x+13)`
for all x that are large (Why did we switch the inequality signs?)
Now, `underset(xtooo)lim(x-7)/(2x+13)=underset(xtooo)lim(1-((7)/(x)))/(-2+((13)/(x)))=(1-0)/(-2+0)=-(1)/(2)`
`underset(xtooo)lim (x+7sinx)/(-2x+13)=-1/2`