# Visually explaining limit properties.

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.

• For example, for the sum rule, if $f(x)$ is getting really close to $L$ and $g(x)$ is getting really close to $M$, it makes sense that $f(x) + g(x)$ is getting really close to $L+M$. Commented Aug 12, 2020 at 14:21

• 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.