Understanding Limits
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UNDERSTANDING LIMITS
Example:
Given the graph of
below, evaluate the following limits.
y f ( x)
(a)
=0 lim f ( x )
lim f ( x)
=DNE lim f ( x )
x 4
(e) lim f ( x )
=2 x
=-1 x 4
(c) lim f ( x)
x 3
=1 x 4
=-∞=DNE lim f ( x )
x 1
(d)
(b)
(f)
(g)
=-∞=DNE
(h)
lim f ( x)
=∞=DNE
(i)
lim f ( x )
x 2
x 2
=DNE lim f ( x )
x 2
Algebraic Limits:
(a)
= -1/4
lim x2
x4 2 x x2
(b)
=6/5 (c)
lim x 3
x2 9 x2 x 6
lim x 2
=5/0=∞=DNE
(d)
=5/0=∞=DNE lim x2
lim
2
=0/1=0
4x 7 x x 3 x 5
lim
2
(g)
=1/4 lim x 4
x 2 x4
=3/2
3x 5 x 2 x 4
x3
x 2
(e)
(f)
x3 x2
Example:
Given the graph of
below, evaluate the following limits.
y f ( x)
(a)
=0 lim f ( x )
lim f ( x)
=DNE lim f ( x )
x 4
(e) lim f ( x )
=2 x
=-1 x 4
(c) lim f ( x)
x 3
=1 x 4
=-∞=DNE lim f ( x )
x 1
(d)
(b)
(f)
(g)
=-∞=DNE
(h)
lim f ( x)
=∞=DNE
(i)
lim f ( x )
x 2
x 2
=DNE lim f ( x )
x 2
Algebraic Limits:
(a)
= -1/4
lim x2
x4 2 x x2
(b)
=6/5 (c)
lim x 3
x2 9 x2 x 6
lim x 2
=5/0=∞=DNE
(d)
=5/0=∞=DNE lim x2
lim
2
=0/1=0
4x 7 x x 3 x 5
lim
2
(g)
=1/4 lim x 4
x 2 x4
=3/2
3x 5 x 2 x 4
x3
x 2
(e)
(f)
x3 x2
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