Understanding the concept of limits and how they are used to define continuity and differentiation in calculus.
Introduction to Limits
Calculus I – Core Concept
A limit describes the behavior of a function (f(x)) as the input (x) approaches a particular value (c).
Formally:
[ \lim_{x\to c} f(x)=L ]
means that the values of (f(x)) can be made arbitrarily close to (L) by taking (x) sufficiently close (but not equal) to (c).
| Application | How Limits Are Used |
|---|---|
| Derivative | (\displaystyle f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}) |
| Integral (Riemann sum) | (\displaystyle \int_a^b f(x),dx = \lim_{n\to\infty}\sum_{i=1}^{n} f(x_i^*)\Delta x) |
| Continuity | (f) is continuous at (c) iff (\displaystyle \lim_{x\to c}f(x)=f(c)) |
| Asymptotes & behavior at infinity | (\displaystyle \lim_{x\to\infty} f(x)) tells us horizontal asymptotes. |
| Series convergence | (\displaystyle \sum_{n=1}^{\infty} a_n) converges when (\lim_{n\to\infty} s_n) (partial sums) exists. |
If both one‑sided limits exist and are equal, the two‑sided limit exists.
For a finite limit (L):
[ \lim_{x\to c} f(x)=L \iff \forall ,\varepsilon>0;\exists,\delta>0; \text{such that}; 0<|x-c|<\delta ;\Longrightarrow; |f(x)-L|<\varepsilon . ]
Key idea: No matter how tight a tolerance we set for the output, we can find a tolerance for the input that guarantees it.
| Type | Typical Form | How to Evaluate |
|---|---|---|
| Finite limit at a finite point | (\displaystyle \lim_{x\to c} f(x)) | Direct substitution (if continuous), factor & cancel, rationalize, use trigonometric identities. |
| Infinite limit | (\displaystyle \lim_{x\to c} f(x |