Question

What is the limit of `((1 + 3 x) / (2 + 4 x^2 + 2 x^4))^3` as x approaches 1?

Answer 1

`y=1/8`

Explanation:

To make the computes easier we can rewrite the expression like

`f(x)=((1+3x)/(2+4x^2+2x^4))^3=((1+3x)/(2*(1+2x^2+x^4)))^3=1/2^3((1+3x)/(x^4+2x^2+1))^3=1/8((1+3x)/((x^2+1)^2))^3`

You have to find

`lim_(x->1)f(x)=lim_(x->1)1/8((1+3x)/((x^2+1)^2))^3=1/8lim_(x->1)((1+3x)/((x^2+1)^2))^3`

Since `f(x)` is continuos in `x=1`, because `(x^2+1)^2!=0 AAx`

to evaluate the limit you could simply substitute `x=1`

`lim_(x->1)f(x)=1/8((1+3(1))/(((1)^2+1)^2))^3=1/8((1+3)/(1+1)^2)^3=`
`=1/8(cancel(4)/cancel(4))^3=1/8*1=1/8`

graph{((1+3x)/(2+4x^2+2x^4))^3 [0.9017, 1.1578, 0.0647, 0.1928]}