Limits
Limits formalize the idea of “approaching” a value.
$$
\lim_{x \to a} f(x)=L
$$
Evaluating Limits
- Determinate form: Direct Substitution
- Indeterminate form ($\frac{0}{0}$) (denominator=0): Needs cancellation (factoring or conjugates)
[!NOTE] Sandwich Theorem (Squeeze Theorem)
If $g(x) \leq f(x) \leq h(x)$ and $\lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L$, then
$$
\lim_{x \to c} f(x) = L
$$
Practice of Squeeze Theorem
[!example]
(a)小问
有三角函数，就找三角函数范围，然后用squeeze theorem
!621x484
Left Hand & Right Hand Limits
Left Hand Limit
$$
\lim_{ x \to 0 ^-} f(x)
$$
Right Hand Limit
$$
\lim_{ x \to 0 ^-} f(x)
$$
Limits and Continuity
[!NOTE] DEF'N
f is continuous at $x_{0}$ means
$$\lim_{ x \to x_{0} }f(x)=f(x_{0})$$
从上面这个定义中，我们可以证明出这个理论：
[!NOTE] DIFF->CTS THEOREM
if f is differentiable at $x_{0}$, then f is continuous at $x_{0}$.
$$
证明过程:
\begin{align}
如果\lim_{ x \to x_{0} }f(x)-f(x_{0})=0那么f就是continous的 \
咱们巧妙的给它上下同乘(x-x_{0})\
=\lim_{ x \to x_{0} } \frac{f(x)-f(x_{0})}{x-x_{0}}(x-x_{0}) \
=f'(x_{0})\times 0 \
所以只要f'(x_{0})是存在的那这个式子就等于0 \
也就是continuous的 \
\end{align}
$$[!NOTE] Types of Discontinuity
- Removable Discontinuity:
- Limit from left & right are equal.
- Jump Discontinuity:
- Limit from left + right exists but are not equal
- So it does not have an overall limit
- Infinite Discontinuity:
- function grows unbounded
- Oscillating Discontinuity
- Oscillation!400x204
[!yellow] Limit (Overall Limit) do not exist on
1. Jump discontinuity
2. Infinite Discontinuity
3. Oscillating Discontinuity
Limit Laws
- Sum: $\lim_{x \to c} (f(x)+g(x)) = L+M$
- Difference: $\lim_{x \to c} (f(x)-g(x)) = L-M$
- Constant Multiple: $\lim_{x \to c} (k f(x)) = kL$
- Product: $\lim_{x \to c} (f(x)g(x)) = LM$
- Quotient: $\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{L}{M}, \, M \neq 0$
- Power: $\lim_{x \to c} [f(x)]^n = L^n$
- Root: $\lim_{x \to c} \sqrt[n]{f(x)} = \sqrt[n]{L}$
The Precise Definition of a Limit
为什么要一个精确的定义？因为“极限接近”这样的模糊定义不足够用于数学证明。
[!NOTE] Precise Definition of a Limit (Epsilon–Delta DEFN)
概念：不管你给个多小的y范围($\epsilon$)，我都能给出一个x范围($\delta$)，里面随便取一个x，对应的f(x)都在这个范围里面for every $\epsilon>0$,there exists a $\delta>0$ such that
$$\begin{align} |f(x)-L|<\epsilon,0<|x-c|<\delta \ \
\end{align}
$$
Practice of Epsilon-Delta DEFN
[!example] eg1
(b)小问!621x484
[!example] eg2
!603x434
Some more examples: Lecture 2.pdf