I need help to resolve this exercise :
Using asymptotic limited development in the neighborhood of + ∞ of the function
$$g(x) = 2x + 3 \sqrt{x^2 -1} $$ Determine an equation of the straight line asymptote to the graph of g in + ∞ and position the representative curve of g with respect to this asymptote.
Sorry for my basic english..
On
the asymptotic will have the form $y=ax+b$
$a= g'(x)\lim_{x\to∞ } =\lim_{x\to∞ }$$2+$ $\frac{3x}{\sqrt {x^2-1}}\ $ =
$ \lim_{x\to∞ }$$2+$ $\frac{3x}{x\sqrt {1-\frac{1}{x^2}}}\ $ = $ \lim_{x\to∞ }$$2+$ $\frac{3}{\sqrt {1-\frac{1}{x^2}}}\ $
$\frac{1}{x^2}$ will be 0 so the limit is
$ \lim_{x\to∞ }$$2+$ $\frac{3}{\sqrt {1}}\ $ $= 2+3$ $=a$
b will be infinity big
So the asymptotic function will be $y=5x$
The question is asking: What does this graph look like for very large $x$?
$$ g(x) = 2x + 3 \sqrt{x^2 -1} \stackrel{\text{subtracting 1 is no biggie for}\;x\sim \infty}{\sim} 2x+3\sqrt{x^2}\\ =2x+3|x|\stackrel{x>0}{=}5x\Rightarrow g(x)\sim 5x $$ or the graph of the function approaches the graph of $y=5x$.
edit: As noted, in my attempt to be cute I may have swept something important under the rug, if instead we had $$ g(x)=2x + 3 \sqrt{x^2 -x} $$ This function would be similar to $5x+b$, since the lower order term in the square root is still dominated, but will add a relatively small but existent perturbation for large $x$.