SAGE : Limits

1. For the function $f(x)=\frac{x^3-1}{x-1}$, find:

  1. $f(1)$
  2. plot $f$
  3. $\lim\limits_{x \to 1}{\frac{x^3-1}{x-1}}
    $

Let's plot the function:


Calculating the limit: $\lim\limits_{x \to 1}{\frac{x^3-1}{x-1}}$


The function looks very much like a parabola, but actually has a hole (or removable discontinuity) at the point $(1, 3)$.


Limits to Infinity

2. Find $\lim\limits_{n \to \infty}{\left(1+\frac{1}{n}\right)^n}$



One-Sided Limits

3. Find $\lim\limits_{x \to 4^+}{\frac{\sqrt{x}-2}{x-4}}$