Prove a limit using the mean value theorem

[Charlotte87]

I am supposed to use the mean-value theorem to show that lim_x→infty(√(x+5)-√(x))=0.

Can anyone help me solving this problem?

I have tried to set up the mean value theorem, but i just do not know how to proceed.

 

[Millennial]

It is rather obvious even without the mean value theorem. Just multiply and divide by the conjugate.

 

[Charlotte87]

I also believe that the limit is obvious, but the exercise here is to prove it using the mean value theorem.

 

[HallsofIvy]

The "mean value theorem" says that, for f continuous on [a, b] and differentiable on (a, b), there exist c between a and b such that
$$\frac{f(b)- f(a)}{b- a}= f'(c)$$.

Take f(x)= x^1/2, b= x+5, a= x. Of course, as x goes to infinity, so do both x and x+5 so a number "between them", c, must also go to infinity. What can you say about f'(c) as c goes to infinity?

 

[Charlotte87]

So, if I write that ((f(b)-f(a))/b-a)=f'(c), and takes the value as you say. I know that f'(x)=1/2sqrt(x). The limit of this when x goes to infinity, is 1 divided by an infinity large number which is 0.

Am I right with my reasoning below: this proves my limit, since lim(f(b)-f(a))=lim(f'(c)*(b-a))=0 since f'(c)=0?