Numerical Limits

Limits

The limit is the foundation of calculus. Informally, $\lim_{x \to a} f(x) = L$ means: as $x$ gets arbitrarily close to $a$, $f(x)$ gets arbitrarily close to $L$.

Why Limits Matter

Limits let us deal with quantities we can't compute directly — like the slope of a curve at a single point, or the sum of infinitely many terms.

Numerical Approximation

We can approximate a limit numerically by evaluating $f$ at points very close to $a$, approaching from both sides:

$\lim_{x \to a} f(x) \approx \frac{f(a - h) + f(a + h)}{2}$

For a smooth function, this central average is very accurate when $h$ is small (e.g., h = 1e-7).

Examples

The third example is famous: $f(0)$ is undefined (division by zero), but the limit exists.

Removable Discontinuities

When $f(a)$ is undefined but the limit exists, the point is called a removable discontinuity. The function $\frac{x^2 - 4}{x - 2} = \frac{(x+2)(x-2)}{x-2} = x + 2$ has a hole at $x = 2$, but the limit is $4$.

Your Task

Implement double limit(double (*f)(double), double x, double h) that approximates the limit of $f$ at $x$ using step size $h$.