Limits
1.2 Limit Laws
Core Laws
Suppose $\lim_{x\to a}f(x)=L$ and $\lim_{x\to a}g(x)=M$.
Theorem
Then:
- $\lim_{x\to a}(f+g)=L+M$
- $\lim_{x\to a}(f-g)=L-M$
- $\lim_{x\to a}(cf)=cL$
- $\lim_{x\to a}(fg)=LM$
- $\lim_{x\to a}(\frac{f}{g})=\frac{L}{M}$ when $M\neq 0$
Proof
The sum law follows from the triangle inequality and the definition of limit. The others are proved similarly with epsilon-delta estimates.
Example
Let $f(x)=x^2+1$ and $g(x)=3x$ at $a=2$.
$$ \lim_{x\to 2}(f(x)+g(x)) = (2^2+1) + 3(2) = 11 $$
Show solution
By the sum law,
$$ \lim_{x\to 2}(f(x)+g(x)) = \lim_{x\to 2} f(x) + \lim_{x\to 2} g(x). $$
Since polynomials are continuous,
$$ \lim_{x\to 2}(x^2+1) = 2^2+1 = 5, \quad \lim_{x\to 2}(3x) = 3(2)=6. $$
Therefore,
$$ \lim_{x\to 2}(f(x)+g(x))=5+6=11. $$
Definition
A rational function is continuous where its denominator is nonzero.
Warning
The quotient law fails if the denominator limit is $0$.
Exercise
Compute $\lim_{x\to 1} \frac{x^2-1}{x-1}$ using limit laws and algebraic simplification.