# Overview definitions limits of functions ```Overview definitions limits of functions
Here is an overview of all the different definitions concerning limits of functions.
The colors are meant to help you see the logic/patterns behind these definitions.
Once you understand them, try if you can reproduce them yourself without
looking.
lim f (x) = L
x→+∞
For all &gt; 0 there exists M &gt; 0 s.t. for all x &gt; M we have
|f (x) − L| &lt; .
lim f (x) = +∞
x→+∞
For all K &gt; 0 there exists M &gt; 0 s.t. for all x &gt; M we have
f (x) &gt; K.
lim f (x) = −∞
x→+∞
For all K &gt; 0 there exists M &gt; 0 s.t. for all x &gt; M we have
f (x) &lt; −K.
lim f (x) = L
x→−∞
For all &gt; 0 there exists M &gt; 0 s.t. for all x &lt; −M we have
|f (x) − L| &lt; .
lim f (x) = +∞
x→−∞
For all K &gt; 0 there exists M &gt; 0 s.t. for all x &lt; −M we have
f (x) &gt; K.
lim f (x) = −∞
x→−∞
For all K &gt; 0 there exists M &gt; 0 s.t. for all x &lt; −M we have
f (x) &lt; −K.
1
lim f (x) = L
x→a
For all &gt; 0 there exists δ &gt; 0 s.t. for all x with 0 &lt; |x − a| &lt; δ
we have |f (x) − L| &lt; .
lim f (x) = +∞
x→a
For all K &gt; 0 there exists δ &gt; 0 s.t. for all x with 0 &lt; |x − a| &lt; δ
we have f (x) &gt; K.
lim f (x) = −∞
x→a
For all K &gt; 0 there exists δ &gt; 0 s.t. for all x with 0 &lt; |x − a| &lt; δ
we have f (x) &lt; −K.
lim f (x) = L
x→a+
For all &gt; 0 there exists δ &gt; 0 s.t. for all x with 0 &lt; x − a &lt; δ we
have |f (x) − L| &lt; .
lim f (x) = +∞
x→a+
For all K &gt; 0 there exists δ &gt; 0 s.t. for all x with 0 &lt; x − a &lt; δ
we have f (x) &gt; K.
lim f (x) = −∞
x→a+
For all K &gt; 0 there exists δ &gt; 0 s.t. for all x with 0 &lt; x − a &lt; δ
we have f (x) &lt; −K.
lim f (x) = L
x→a−
For all &gt; 0 there exists δ &gt; 0 s.t. for all x with 0 &lt; −(x − a) &lt; δ
we have |f (x) − L| &lt; .
lim f (x) = +∞
x→a−
For all K &gt; 0 there exists δ &gt; 0 s.t. for all x with 0 &lt; −(x − a) &lt; δ
we have f (x) &gt; K.
2
lim f (x) = −∞
x→a−
For all K &gt; 0 there exists δ &gt; 0 s.t. for all x with 0 &lt; −(x − a) &lt; δ
we have f (x) &lt; −K.
3
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