Confusion related to definition of a derivative at a point in its domain of definition.

Question

I have just completed relations and functions and now I am studying limits and differentiability.As a beginner help me to clear my doubts.

My textbook defines limits in this way:-

$$\lim_{ x\to a } = l $$

I grasp it in this way that as x approaches to take value 'a' then function tends to become 'l'. I know that as x tends to become 'a' from both sides left hand side as well as right hand side then function also tends to become 'l' from both sides. I also know that limit of a function may or may not be equal to the value of the function at that point in domain.

Now I have basically two confusions whether the algebra of limits is a law or a definition.I can verify the properties of algebra of limits by considering some real valued functions but I am unable to prove it.

My next doubt is related to definition of a derivative of a function at a point in its domain which is stated in my textbook in this way:-

$$\lim_{h\to 0} \frac{f(a+h)-f(a)}{h}$$

Now I know that as 'h' tends to become 0 then the expression after the limit has some finite limit but I can't perceive it in the notion of limits. I know it's geometric interpretation that this is the slope of the tangent to the curve at point 'a' .

I am very much confused that how I can interpret it in the way I do for limits. Please tell me if I am not able to convey my problem. Please edit it if there is any problem in math Jax.

Answer 1

The idea of the definition of derivatives is: that if you have two points on the graph of function $f$ so that your two points are $(x_1, f(x_1)) $ and $(x_2, f(x_2))$ then the slope between those lines is $\frac {f(x_2) - f(x_1)}{x_2 - x_1}$.

Now the idea of a derivative at a point $x$ is if we assume $f$ is a "nice" at that point (continuous, not jaggy,etc) so that it has a well define tangent line at $x$ the derivative at the point $x$ will be the slope of the tangent line. (If the function, $f$, is "nice" then every, or at least several, different points $x$ will each have their own tangent lines with different slopes.)

Now to measure the slope of the tangent line we have to take two points of the tangent line. But we only know one point of the tangent line: $(x,f(x))$.

So what we do is we take a point that is on the function that is very near $x$, say $x_k$ and figure that if $x_k$ is very close to $x$, then the slope of the line between $(x,f(x))$ and $(x_k, f(x_k))$ will be pretty close to the slope of the tangent line and the closer $x_k$ is to $x$ the closer the slope of the line will be to the slope of the tangent line.

The way I like to picture this is the we have the point $(x, f(x))$ fixed. And we slide $x_k$ closer to $x$ and so what happens as $f(x_k)$ gets closer to $x$ and how the slope gets close to the slope of the tangent line. Fix $x$ and slide $x_k$.....

So assuming this function is so that everything is "nice" we will figure

$$\text{slope of tangent line at }(x,f(x))=\lim\limits_{x_k\to x}\text{slope of the line }(x,f(x))\text{ to }(x_k, f(x_k)) = \lim\limits_{x_k\to x}\frac {f(x_k) - f(x)}{x_k- x}$$

(I should note that it doesn't matter if $x_k > x$ or $x_k < x$ of if $f(x_k)$ is to the right or left of $f(x_k)$. We can calculate the slope from either point as $\frac {y_2 - y_1}{x_2 - x_1} = \frac {y_1 - y_2}{x_1 - x_2}$.)

For reasons that will become clear with practice, it's more insightful and easier and practical, to instead of using $x_k$ to use $x_k = (x + h)$ where $h = x_k - x$, the (directional) distance between $x_k$ and $x$. (Note: $h$ might be negative. It doesn't actually matter.)

So if the function is nice:

Slope of tangent line = limit of slope of tangent line between $(x, f(x))$ and $(x+h, f(x+h))$ as $x+h\to x$ = $\lim\limits_{x+h \to x} \frac{f(x+h) - f(x)}{(x+h)-x} = \lim\limits_{h\to 0}\frac{f(x+h) -f(x)}h$.

And that should answer your question.

Answer 2

Yes the algebra of limits can be proved and you may read it in any standard first year undergraduate calculus book like Apostol.And in second part, it is not necessary that limit may exist, it can diverge , in case it exists , we say it is detivative at that point. Geometrically, notion of limit can be taken as a considering the value to which tangents drawn in neighbourhood points of x tend to converge

Answer 3

The following is the definition of the derivative function provided the limit of the difference quotient as $h\to 0$ exists, if you wish to find the slope at a point say $x=a$ then just put $a$ in for $x$. $$f'(x)\equiv\ \lim_{h\to 0}\underbrace{\dfrac{f(x+h)-f(x)}{h}}_{\text{difference quotient}}$$

Now coming to the laws of limits yes they can be proved, these proofs make use of the formal $\epsilon -\delta$ definition of the limit. This link: Proof of various Limit laws contains what you're looking for.