I have been in a deep confusion for about a month over the topic of limits! According to our book, the limit at $a$ is the value being approached by a function $f(x)$ as $x$ approaches $a$.
I have a doubt that in real number line we can never ever reach the closest value to $a$ because always a more closer value will exist.
Now when talking about our methods for calculating limit, what method comes in our mind when we have to calculate where the value approaches? Let's say we have to find what value $f(x)$ approaches when $x$ moves from $[0,a)$.
So we calculate the value of $f(x)$ at $x=0$, say $4$, and then at $x\to a$.
But the problem is that we don't know the value $x\to a$, so we say the value is $a-h$ where $h\to 0$ and calculate the value of $f(a-h)$, say $5-h$.
Now here is where my doubt starts! In the final step we put the value of $h=0$ say in that it is an infinitesimal quantity.
My doubt is that $h$ was tending to $0$ means that it was never equal to zero maybe it is infinitesimally small, not a stationary value, not an imaginable value, but we know for sure that it is not equal to $0$. Maybe it is the point closest to zero, but it is not equal to zero and when we use the result $5-h=5$ we are actually making an error which is tending to zero. Maybe the error is very small, but still, there is some error in that we cannot calculate it but we can see that there is this infinitesimal error present.
That means we don't get the exact limiting value or last value of $f(x)$.
$x$ belongs to $[0,a)$ but a value approximate to infinitesimal? Isn't it right! We get an approximated value?
Best Answer
Well, in a sense, you're right. When they say that the limit of $f(x)$ at $x=a$ is $L$ it doesn't necessarily mean that $f(a)=L$. Actually, the nice idea behind limits is that you can talk about the limit of a function even if the function is not defined at that value. This is a very powerful idea that later enables us to talk about derivatives as you possibly know.
For example $\displaystyle \lim_{x\rightarrow 0}\frac{\sin(x)}{x}=1$ but the value of $\displaystyle \frac{\sin(x)}{x}$ is not defined at $x=0$. If you graph it on wolframalpha, you'll see that this means 'as we approach $x=0$ the value of $\displaystyle \frac{\sin(x)}{x}$ approaches 1'. We never claim that these two are equal! We just claim that the value of $f(x) = \displaystyle \frac{\sin(x)}{x}$ can become arbitrarily close to $1$ provided that we let $x$ be close enough to $0$ .
When we say that the limit of $f(x)$ at $x=a$ is $L$, we are claiming that we can make $f(x)$ arbitrarily close to $L$ provided that we take $x$ close enough to $a$. That's all.