limits and continuity

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LIMITS AND CONTINUITY

1. The general of limit
x2 4
. Examine the demeanor of f (x) as x approaches 2.
Example 1.1. Let f (x) =
x2
Solution. Let us compute some values of f (x) notwithstanding x close to 2, as in the tables in the regions of the dead.

We see from the rst level that f (x) is getting closer and closer to 4 taken in the character of x approaches 2 from
the left indirect. We express this by saying that “the circumscribe of f (x) as x approaches 2 from left is
4”, and
lim f (x) = 4.

x2

Similarly, through looking at the second table, we argue that “the limit of f (x) while x approaches 2 from
right is 4”, and
lim f (x) = 4.

x2+

We assemble lim f (x) and lim f (x) human being-sided limits. Since the two unit-sided limits of f (x) are

+
x2

x2

the similar, we can say that “the restrain of f (x) as x approaches 2 is 4”, and
lim f (x) = 4.

x2

Note that since x2 4 = (x 2)(x + 2), we have power to
x2 4
x2 x 2
(x 2)(x + 2)
= lim
x2
x2
= lim (x + 2) = 4,

lim f (x) = lim

x2

x2

whither we can cancel the factors of (x 2) from that time in the limit as x 2, x is terminate to 2, but
x = 2, in such a manner that x 2 = 0. Below, nd the graph of f (x), from what one. it is also clear that
limx2 f (x) = 4.
1

2

Example 1.2. Let g(x) =

LIMITS AND CONTINUITY

x2 5
. Examine the behavior of g(x) as x approaches 2.
x2

Solution. Based in successi~ the graph and tables of approximate business values shown below,

observe that in the same proportion that x gets closer and closer to 2 from the left, g(x) increases on the outside of bound and
as x gets closer and closer to 2 from the left, g(x) decreases out of bound. We express this
situation ~ the agency of saying that the limit of g(x) because x approaches 2 from the left is , and g(x) while x
approaches 2 from the up~ is and
lim g(x) = ,

x2

lim g(x) = .

x2+

Since in that place is no common value for the single in kind-sided limits of g(x), we rehearse that the limit of g(x)
because x approaches 2 does not exists and
lim g(x) does not exits.

x2

Example 1.3. Use...

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