Question

How do you find the limit of `sqrt(x+e^(4x))/(e^(2x)+x)` as `x->oo`?

Answer 1

`lim_(x->oo) sqrt(x+e^(4x))/(e^(2x)+x) =1`

Explanation:

Bring the denominator under the square root:

`f(x) = sqrt(x+e^(4x))/(e^(2x)+x) = sqrt(((x+e^(4x))/(e^(2x)+x)^2)`

Expand:

`f(x) = sqrt(((x+e^(4x))/(e^(4x)+2xe^(2x)+x^2))`

Divide by `e^(4x)` above and below:

`f(x) = sqrt(((xe^(-4x)+1)/(1+2xe^(-2x)+x^2e^(-4x)))`

As for every `n` and `alpha>0`: `lim_(x->oo) x^n e^(-alpha x) = 0`

`lim_(x->oo) f(x) = lim_(y->1) sqrt(y) = 1`