Finding derivative using limit definition

Why can you find a derivative using its limit definition like so:

I get you can simplify $$\displaystyle \lim_{h\to 0} \frac{(x+h)^2-x^2}{h} $$

to $$\displaystyle \lim_{h\to 0} \frac{2xh + h^2}{h} $$

But why are you allowed to simplify further to: $$\displaystyle \lim_{h\to 0} {2x + h} $$

You can't use the quotient limit law and take the limit of the numerator divided by the limit of the denominator because the limit of the denominator is zero, right? But why are you allowed to divide h first and simplify before doing the limit? Can you always divide h out of the expression and take the limit of the new expression? If so, why?

Solution 1:

I am not sure how much background in limits you have. So, I will try to keep it simple but try to introduce you to some rigour.

To understand your question, we must first look into the expression "$\lim\limits_{x \rightarrow a} \langle \text{some expression in }x \rangle$". For the time being, we will not be concerned about the expression, but only will look at the meaning of "taking limits". The first question that comes to mind when we look at what is written is, "What do we mean by $x \rightarrow a$?"

To put it simply, it means that we can come "as close as we wish" to the point $a$ without actually touching it, and then look at the value of the expression as we keep on coming "close". Again, here, the question is, "how close is close enough?" And that depends on us! We can start by saying that let us be at a "distance" of at most $1$ from point $a$ and let us call this close. Then we may become more demanding and say that alright, now let us get a bit closer to $a$, say at a distance of atmost $\frac{1}{2}$. And we can keep on doing this.

Another thing that we must address is that what do we mean by "distance"? In the real case (which you have asked in your question), the absolute value of the difference gives the distance (if you are unsure about this, try to draw some diagrams and convince yourself). Now, to write all this intuition mathematically, we will shift to symbols.

For simplicity, I will replace "$\langle \text{some expression in } x \rangle$" by $f \left( x \right)$. We will say that $\lim\limits_{x \rightarrow a} f \left( x \right) = l$, if whenever we come "close enough" to $a$, the value of the expression $f \left( x \right)$ is "close enough" to $l$. Now that we have intuition of what it means to be close enough, we will try to write it mathematically.

First, let is fix a permissible "closeness" of the expression $f \left( x \right)$ to $l$, and call it say $\varepsilon$. Now, we would like to know at what degree of "closeness" to $a$, does $f \left( x \right)$ really come close to $l$? That is we are asking whether such a permissible closeness, $\delta$, exists? So, the definition of the limit becomes

$$\forall \varepsilon > 0, \exists \delta > 0 \text{ such that } \forall x \text{ with } 0 < \left| x - a \right| < \delta, \text{ we have } \left| f \left( x \right) - l \right| < \varepsilon.$$

I would like you to take some time to digest this transition from the intuition in the English language to the concrete definition in Mathematical symbols.

Now, you observe that we have made the condition $0 < \left| x - a \right| < \delta$, which translates to "$x$ can come close enough to $a$ without actually touching it". And this is the precise reason why In your question, we don't take $h = 0$. This is also the reason that you can make all manipulations in your expression $f \left( x \right)$, as long as it makes sense mathematically (i.e., you do not perform some invalid operations such as dividing by $0$ or cancelling off zeros from the numerator and denominator).

I hope this answers your question.