Visually explaining limit properties.

Question

I was doing the start of the calculus section from khan academy. I couldn't properly understand the limit properties.

Limit properties: https://miro.medium.com/max/565/0*9pLbmbpHrva-LtMe.png

I searched a lot for a graphical explanation of these properties and how the limit work with composite functions, but couldn't find a visual explanation telling me why these properties are true.

This is quite a basic thing but I really think a visual explanation will help me understand it better.

Answer 1

These properties come from the following basic facts:

• When $\lim_{x\to c} f(x)=L$ and $\lim_{x\to c}g(x)=M$ then $\bigl(f(x),g(x)\bigr)$ is near $(L,M)$ when $x$ is near $c$.
• The arithmetic operations $+$, $-$, $*$, $:$ (when defined) are continuous. This means that when $a'$ is near $a$ and $b'$ is near $b$, then $a'\cdot b'$ is near $a\cdot b$, for all four operations.