Here are some techniques for solving math problems.
Adding new term while keep the equation true
aab=b−c+c=cb×ac
Manipulating fractions
1−n1n−1nn(n+1)1=nn−1=n−1n−1+n−11=1+n−11=n1−n+11
Example: Limit of (1−1/n)n as n→∞
(1−n1)n=(nn−1)n=(n−1n1)n=(1+n−111)n=(1+n−111)n−1⋅1+n−111
Recall that:
e:=n→∞lim(1+n1)n
As n→∞, the left term goes to e1 and the right term goes to 1. So:
n→∞lim(1−n1)n=e1
Absolute values
For a,b∈R:
a∣=∣−a∣ab≤∣a∣∣b∣a2=∣a∣−∣a∣≤a≤∣a∣∀r>0,∣a∣≤r⟺−r≤a≤r
From the triangle inequality, for a,b,c∈R:
∣a+b∣∣a−b∣∣a−b∣≤∣a∣+∣b∣≤∣a∣+∣b∣≤∣a−c∣+∣c−b∣