Question

Evaluate the limit? : `lim_(x rarr oo)(3x+1)/(|x|+2)`

Answer 1

`lim_(x rarr oo)(3x+1)/(|x|+2) = 3`

Explanation:

As `x rarr oo => x>0` so:
`\ \ \ \ \ lim_(x rarr oo)(3x+1)/(|x|+2) = lim_(x rarr oo)(3x+1)/(x+2)`
`:. lim_(x rarr oo)(3x+1)/(|x|+2) = lim_(x rarr oo)(3x+1)/(x+2)*(1/x)/(1/x)`
`:. lim_(x rarr oo)(3x+1)/(|x|+2) = lim_(x rarr oo)((3x+1)/x)/((x+2)/x)`
`:. lim_(x rarr oo)(3x+1)/(|x|+2) = lim_(x rarr oo)(3+1/x)/(1+2/x)`

As `x rarr 8 => 1/x rarr 0`, Hence
`\ \ \ \ \ lim_(x rarr oo)(3x+1)/(|x|+2) = (3+0)/(1+0) = 3`

We can verify this by looking at the graph of `y=(3x+1)/(|x|+2)`

graph{(3x+1)/(|x|+2) [-5, 20, -3.5, 3.5]}