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
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lim f (x) = L
x→+∞
For all > 0 there exists M > 0 s.t. for all x > M we have
|f (x) − L| < .
lim f (x) = +∞
x→+∞
For all K > 0 there exists M > 0 s.t. for all x > M we have
f (x) > K.
lim f (x) = −∞
x→+∞
For all K > 0 there exists M > 0 s.t. for all x > M we have
f (x) < −K.
lim f (x) = L
x→−∞
For all > 0 there exists M > 0 s.t. for all x < −M we have
|f (x) − L| < .
lim f (x) = +∞
x→−∞
For all K > 0 there exists M > 0 s.t. for all x < −M we have
f (x) > K.
lim f (x) = −∞
x→−∞
For all K > 0 there exists M > 0 s.t. for all x < −M we have
f (x) < −K.
1
lim f (x) = L
x→a
For all > 0 there exists δ > 0 s.t. for all x with 0 < |x − a| < δ
we have |f (x) − L| < .
lim f (x) = +∞
x→a
For all K > 0 there exists δ > 0 s.t. for all x with 0 < |x − a| < δ
we have f (x) > K.
lim f (x) = −∞
x→a
For all K > 0 there exists δ > 0 s.t. for all x with 0 < |x − a| < δ
we have f (x) < −K.
lim f (x) = L
x→a+
For all > 0 there exists δ > 0 s.t. for all x with 0 < x − a < δ we
have |f (x) − L| < .
lim f (x) = +∞
x→a+
For all K > 0 there exists δ > 0 s.t. for all x with 0 < x − a < δ
we have f (x) > K.
lim f (x) = −∞
x→a+
For all K > 0 there exists δ > 0 s.t. for all x with 0 < x − a < δ
we have f (x) < −K.
lim f (x) = L
x→a−
For all > 0 there exists δ > 0 s.t. for all x with 0 < −(x − a) < δ
we have |f (x) − L| < .
lim f (x) = +∞
x→a−
For all K > 0 there exists δ > 0 s.t. for all x with 0 < −(x − a) < δ
we have f (x) > K.
2
lim f (x) = −∞
x→a−
For all K > 0 there exists δ > 0 s.t. for all x with 0 < −(x − a) < δ
we have f (x) < −K.
3