- Understand how to write the limit of a function using the correct symbols and estimate limits by examining tables and graphs
- Understand one-sided limits (approaching a point from only one direction) and how they relate to two-sided limits
- Understand and use the proper notation for infinite limits and define vertical asymptotes
The number [latex]e[/latex] is a constant that, like [latex]\pi[/latex], is irrational (it cannot be represented as a ratio of integers) and transcendental (it is not the root of a non-zero polynomial of finite degree with rational coefficients). It was first discovered by the Swiss mathematician Jacob Bernoulli in 1683 while studying compound interest. It has many real-world applications including probability theory and exponential growth and decay. Below is an approximation of [latex]e[/latex] to [latex]20[/latex] decimal places.
[latex]e \approx 2.71828182845904523536[/latex]
This number is perhaps best defined by a limit. Consider the function [latex]f(x)=(1+x)^{\frac{1}{x}}[/latex]. We cannot compute [latex]f(0)[/latex], but we can see what happens with values of [latex]x[/latex] close to zero.