Limits, Continuity & Composite Functions
Why you need to know
- They are the foundation of Calculus (without them, derivatives and integrals cannot be defined).
- They help describe how functions behave when approaching a value (like zooming in infinitely).
- They ensure that real-world models (motion, growth, physics, economics) are smooth and predictable.
How it helps
- Limits → define instantaneous speed, slopes, derivatives.
- Continuity → ensures no sudden breaks in functions (useful in physics, signals, graphs).
- Composite functions → allow combining simple functions into complex ones (used in modeling real problems).
Where it came from
- Originated in the 17th century with early Calculus.
- Formal definitions came in the 19th century to remove vagueness in “approaching values.”
Why it was invented
- To rigorously explain instantaneous change and avoid contradictions in calculus.
- To make math precise when dealing with infinity or very small values.
Who invented
- Limits & Continuity → ideas by Newton & Leibniz (1600s), formalized by Cauchy and Weierstrass (1800s).
- Composite functions → used since early function theory (Euler, 1700s).
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Core Ideas (Simple Terms)
1. Limit
A limit tells us the value a function gets closer to, as input gets closer to some point. Example:
lim_{x tends to 2} (x^2) = 4
(As x gets near 2, x^2 gets near 4).
2. Composite Function
Joining two functions into one:
(f o g)(x) = f(g(x))
Example: if f(x) = x^2, g(x) = x+1, then (f o g)(x) = (x+1)^2.
3. Continuity
A function is continuous at x = a if:
lim_{x tends to a} f(x) = f(a)
Meaning → no jumps, no holes, no breaks at that point.
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✅ In short:
- Limit → approaching value.
- Composite → combining functions.
- Continuity → smooth, unbroken curve.