SOLUTION: Having trouble understanding how limits questions like this work, could you please explain how to solve this type of question? find lim h->0 (f(x+h)-f(x))/h for f(x)=sqrt(7+2x)

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Question 600349: Having trouble understanding how limits questions like this work, could you please explain how to solve this type of question?
find lim h->0 (f(x+h)-f(x))/h for f(x)=sqrt(7+2x)
Answer by solver91311(24713) (Show Source):
You can put this solution on YOUR website!

As ugly as this thing looks, the solution is both simple and elegant.

The trick is to multiply by 1, but in a special form. Remember the factorization of the difference of two squares? Go back the other way: The product of a conjugate pair of binomials, i.e. is the difference of two squares, . So what? Watch:

Multiply by 1 in the form of

Now create the difference of two squares in the numerator:

Collect like terms in the numerator:

Get rid of the factor of that is now in both numerator and denominator.

Now take the limit

making the other just go away.

Collect terms in the denominator

And finally, lose the 2 that is in both numerator and denominator:

Done.

John

My calculator said it, I believe it, that settles it