Limits and Continuity
Limits formalize the idea of approaching a value. Continuity requires that the limit equals the function value. These concepts underlie all of calculus and analysis.
Key Points
- The epsilon-delta definition gives a rigorous foundation for limits.
- A function is continuous at a point if the limit exists and equals the function value.
- Continuity is preserved under sums, products, and compositions (where defined).
Formulas
Limit definition
$$\lim_{x \to a} f(x) = L \iff \forall \varepsilon > 0, \exists \delta > 0 : 0 < |x-a| < \delta \implies |f(x)-L| < \varepsilon$$
Continuity
$$f \text{ is continuous at } a \iff \lim_{x \to a} f(x) = f(a)$$
Code Example
import numpy as np
def f(x):
return np.sin(x) / x
x = np.linspace(-0.01, 0.01, 1000)
# f(x) approaches 1 as x -> 0
print(f(x).mean()) # ~ 1.0