Limits: The fundamental idea of Calculus
Before deriving or integrating, we need to understand what happens to a function as we approach a point. That is exactly what limits study.
The Limit: Formal Definition & Intuition
Limit of a function
We say that $\displaystyle\lim_{x \to a} f(x) = L$ if for every $\varepsilon > 0$ there exists a $\delta > 0$ such that:
In words: we can make $f(x)$ as close to $L$ as we want, simply by choosing $x$ sufficiently close (but not equal) to $a$.
We don't care what exactly happens at $x = a$, only what happens near that point. The function can be undefined at $a$ and the limit can still exist.
Geometric Intuition
Imagine walking down a road and asking yourself: "Where am I heading?". It doesn't matter if you ever exactly reach that destination — what matters is the direction. Limits in mathematics do exactly that: they ask what value a function approaches as the input variable approaches a certain point.
Function: $f(x) = \dfrac{x^2 - 4}{x - 2}$
The notation we use to express a limit is:
"The limit of $f(x)$ as $x$ approaches $a$ equals $L$"
Numerical Approach & One-sided Limits
If we try to evaluate $f(2)$ directly, we get a division by zero ($\frac{0}{0}$), which is undefined. We cannot simply plug in $x=2$.
However, if we evaluate the function at values very close to 2, we notice a clear pattern:
| $x$ | $1.9$ | $1.99$ | $1.999$ | $2.0$ | $2.001$ | $2.01$ | $2.1$ |
|---|---|---|---|---|---|---|---|
| $f(x)$ | $3.9$ | $3.99$ | $3.999$ | Undef. | $4.001$ | $4.01$ | $4.1$ |
Notice how as $x$ gets closer and closer to $2$, the resulting $y$ values converge towards $4$ from both sides.
This is the essence of a limit: we don't care what happens exactly at $x=2$, but rather what destination the function is heading towards.
One-sided limits
For a limit to exist, the limits from the left and from the right must be equal:
Limit from the left
Limit from the right
The limit exists $\iff L^- = L^+$