Question

Lim x - (1/2)(ln (1+ e^2x) + (1/2)(ln (2)) when x---> +(infinie) ?

lim x - (1/2)(ln (1+ e^2x) + (1/2)(ln (2)) when x---> +(infinie)

Answer 1

`lim_(x->oo)(oo-1/2)ln(1+e^(2x))+1/2ln(2)=(oo-1/2)ln(1+e^(2*oo))+1/2ln(2)=(oo)ln(oo)+1/2ln(2)=oo`

Explanation:

We want

`lim_(x->oo)(x-1/2)ln(1+e^(2x))+1/2ln(2)`

Generally, the first thing we should do with these limits is plug in the value we're approaching to see if there's even any need for further simplification. In this case, we'll plug in `oo:`

`lim_(x->oo)(oo-1/2)ln(1+e^(2x))+1/2ln(2)=(oo-1/2)ln(1+e^(2*oo))+1/2ln(2)=(oo)ln(oo)+1/2ln(2)=(oo)(oo)=oo`

We rewrite `(oo-1/2)` as `oo.` Subtracting a small quantity from a very large number still yields the very large number.

Moreover, since `e^(2*oo)=oo, ln(1+e^(2*oo))=ln(1+oo)=ln(oo)=oo`

So, our limit goes to `oo.`