Why limits exist at all
A limit answers one question: as an input gets closer and closer to some value, what does the output get closer and closer to? Not what the output is at that value — what it's approaching. That distinction is the entire reason limits exist as their own concept instead of just being "plug in the number."
Take f(x) = (x^2 - 1) / (x - 1). Plug in x = 1 and you get 0/0 —
undefined. But factor the numerator: (x-1)(x+1) / (x-1), and for every
x except exactly 1, that simplifies to x + 1. As x creeps toward 1
from either side, f(x) creeps toward 2. The function has a hole at
x = 1, but the limit as x approaches 1 is still 2.
Reading the notation
lim(x→a) f(x) = L reads as "the limit of f(x), as x approaches a, is L."
Three separate pieces: the function, the value x is approaching, and the
value the output settles near. None of them require f(a) to actually
exist or equal L.
One-sided limits
Sometimes the answer depends on which direction you approach from. The
left-hand limit (x→a⁻) and right-hand limit (x→a⁺) can disagree — a
step function is the clearest example, jumping from one value to another
at a boundary. The two-sided limit only exists when both sides agree.
Where this actually gets used
Limits aren't just a hoop before derivatives — the derivative itself is defined as a limit (the slope of a secant line as the two points collapse into one). Continuity is defined in terms of limits. Every integral is a limit of sums. Once limits click, the rest of calculus reads as consequences of this one idea rather than a pile of separate rules to memorize.