Intuitive Approach
Suppose the function f(x) = x
+ 7. And let's say x = 1. Now let's increase x from 1 to 5 in
steps of 1.
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x = 1
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f(1) = 8
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x = 2
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f(2) = 9
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x = 3
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f(3) = 10
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x = 4
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f(4) = 11
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x = 5
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f(5) = 12
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For values of x just less than 5, that
is as x is close to 5, f(x) is close to
12.
Also, for values of x that are
approximately equal to 5, f(x) is approximately
equal to 12.
As x
approaches 5 then f(x) approaches 12. 12 is the limit
as x approaches 5.
The word approaches is represented bythe symbol →. So the previous
statement would be written
as x → 5, f(x) → 12.
Suppose we change this function to
In this case the
function is identical except at x = 5. Here it “jumps”
to 20. Yet we still say the limit of this new function as x
approaches 5 is still 12. In fact if there was no value for f(x) at x
= 5, the limit still exists and is 12.
All we care
about is how case f(x) behaves near
x=5.
The notation used to express a limit
is:
We read this like so: the limit
of f-of-x as x approaches c is l.
The notation
“xc”
means that x approaches c from either side of c. In other words, the limit exists and is equal to l
in both cases; as x increases toward c and as x decreases toward c.
The following graph illustrates the
meaning of a limit.
This graph shows a function f(x) in
red. If we start at point p1 and move x toward x = c,
p1 moves along the curve towards p2. f(x)
moves toward y = l, which is the value of f(x) at x = c. This
behavior is shown by the blue arrows.
Likewise if we move x starting at point
p3 toward x = c, p3 moves along the curve
toward p2. f(x) moves toward y = l, which again, is the value of
f(x) at x = c. This behavior is shown by green
arrows.
The following graph illustrates a
jump at x = c.
Notice
at x = c, if(c) is not on the curve, but above it creating a hole on the curve at p2.
The curve approaches l as x decreases and as x increases toward c.
With limits we do not concern ourselves with what happens at x = c, but we concern ourselves with the values the function
approaches as x becomes nearer and nearer to c. If these values are equal then the limit exists at x = c.
Here is an example of a function where
the limit does not exist at x = c.
This function is f(x) = 5 for x <
c, f(x) = -5 for x > c, at x=c f(x) does not exist. A hole is shown by the
circles at the endpoints of the branches at x = c.
As x approaches c from x < c, f(x)
remains 5. As x approaches c from x > c, f(x) remains -5. p1
and p2 represent these limits on each side of x=c and
their y coordinates are different.
In this case the
limit of f(x) as x approaches c does not exist.
The limits from each side of x=c are called
one-sided limits. One-sided limits at x = c must exist and be equal for the
limit to exist.
A special notation is used for one-sided
limits that uses up and down arrows.
The
limit of f(x) as x increases toward c is l.
The
limit of f(x) as x decreases toward c is l.
For the previous example we would
write:
and
but
does not exist.
Here is another example where the limit
does not exist at x = c.
This example is the function f(x) =
1/x, and c = 0.
Recall that f(x) is a hyperbola with
the Y and X axes as asymptotes.
In blue we
see that as x approaches zero f(x) becomes more and more negative
with no limit, that is, it decreases with no bound.
Using limit notation we express this as:
And in green
as x approaches zero f(x) becomes more and more positive with no
limit, that is, increases with no bound.
Using limits this is expressed as:
Since here is no limit either way we approach 0 therefore the
limit at x = 0 does not exist.
Now as x increases with no limit, f(x) decreases toward 0. The
X axis is the asymptote and the Y coordinate of the X axis is the limit, 0, for f(x).
Mathematically we express this like so:
Likewise as x decreases (becomes more negative) with no limit, f(x) again
approaches 0. The negative X axis is the asymptote whose Y coordinate is the
limit of f(x).
Mathematically:
As x approaches zero there is no limit for f(x).
The Y axis is
the asymptote for f(x).
f(x) never crosses the X axis but gets closer and
closer to it, approaching 0.
The word infinity,
represented by the symbol ∞, is not a number. It is conceptually
incorrect to think of it as a number.
This equation:
is read like so:
as x increases toward zero, f(x) decreases negatively with no limit.
Having said this, it is common to hear this equation read this way: the limit of f(x) as x approaches zero is negative infinity.
Infinity is a concept, it is used to describe a nonempty set that has no cardinality.
The meaning we seek is "there is no number to express this limit," that is, there is no limit, or the limit simply
does not exist. We stretch the meaning in equations involving limits. The limit equation expresses an equality and because of the equal sign,
we usually think of the strict equality of two numbers. With limits we don't think of equality but we think of nearness. We then place
the concept, ∞, in a position usually occupied by a number. We're overloading the concept of strict equality in this case
with a shorthand notation to express "with no bound." And this leads to the conceptual error that ∞ is a number.
Make your best attempts to read these equations involving ∞ correctly, doing so will pay off in higher level Mathematics courses.
Example
Using this graph for f(x):
as x increases
toward 4, f(x) increases toward the limit 7
as x decreases
toward 4, f(x) increases toward the limit 4
there is no limit,
since the previous one sided limits differ
as x increases
toward 9, f(x) increases toward the limit 6
as x decreases
toward 9, f(x) increases toward the limit 7
there is no limit, since the previous one sided limits differ
f(x)
is not defined in this interval ( ∞
, -7), there is no
limit
f(x)
becomes more negative as x approaches -7, the asymptote,
there is no limit.
there
is no limit since the previous one sided limits do not exist
Here are some results regarding
limits.
If f(x) and g(x) are functions where:
“The
limit of a constant is that constant.”
“The
limit of a constant multiplied by a function is that constant multiplied by the limit of the function.”
“The
limit of the sum is the sum of the limits.”
“The
limit of the product is the product of the limits.”/P>
“The
limit of the quotient is the quotient of the limits.”
Later we'll call f(x) and g(x) "continuous" functions.
Examples:
lim ( x + 6) = lim x + lim 6
lim( 2 – x) = lim 2 - lim
x = 2 – lim x
lim 3x = lim 3 lim x = 3 lim x
lim x/5 = (lim x) / (lim 5) =
(lim x) / 5
Algebraic examples:
Find the limits.
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1)
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As x approaches 5, (x) approaches 5. The limit is 5.
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2)
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as x approaches 5, x2 approaches 25, this limit is
the sum of these individual limits: 25 + 5 = 30.
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3)
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This limit does not exist. As x approaches 0, 1/x becomes
larger and larger with no bound.
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4)
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As x approaches c, x – c approaches 0, and as in the
previous example 1/(x-c) becomes larger and larger with no bound.
Therefore the limit does not exist.
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5)
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At x = 3 this ratio is indeterminate. But we are not concerned
with x = 3, but with x very close to 3. So,
as long as x ≠
3 then we can divide (x – 3) into x2
– x - 6 and get (x + 2).
(by
factoring the numerator to (x-3)(x+2) and canceling (x-3)).
Now
as x approaches 3, this ratio approaches
(3
+ 2) = 5.
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6)
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As x → 4,
(x + 4) → 0, but we are not concerned at x = ―4.
For x ≠
―4, we can simplify
this ratio to 1, and we have
The
limit is therefore 1.
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7)
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At x = 5 this ratio is 0/0 is indeterminate. But for all x
near 5 we can simplify this ratio to 1/(x-5). And we saw
in example 4, that this limit does not exist.
(note: (x-5)(x-5) = x2 -10x + 25)
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8)
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The limit argument is a constant therefore any change in x has
no change in 15. The limit is therefore 15.
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9)
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Be careful here. This example refers to the limit of 6y
NOT the limit of (6y + 12). Notice that the grouping
symbols are missing in this example. 12 is NOT part of the limit expression. We add 12 to the limit we
of 6y. f(x) = 6y. f(x) means a function
whose independent variable is 'x'. Unless otherwise stated, 6y is not affected by
the change in x. 6y is constant so its limit is 6y. But we must include 12 so this whole
expression becomes 6y + 12. The limit is 6y + 12.
Yes, we normally think of y = f(x). But do not assume it is
unless this relationship is made explicit. In this case it is not
defined as such.
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9a)
#9, and let
y = 2x + 1
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Here we need to substitute (2x + 1) for y in the limit argument to get
6(2x + 1) = (12x + 6).
Now as x → 15 then (12x +6) → (12 ⋅ 15 + 6) = 186
So this limit is 186 + 12 = 198.
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10)
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At x = 3, f(x) equals -15 and 15, so f(x) is NOT
defined at x = 3; this function “jumps from -15 to +15 at x
= 3. This is ok because we're concerned only with x close
to 3.
In this case the left sided limit (x < 3, approaching 3
from the left) is (-5)(3) = -15. The right sided limit is
(5)(3)=15. For a the limit to exist it must be the same from both
sides. Here it is not, -15 ≠
15, so the limit does not exist.
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11)
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For x < 0, f(x) is negative; for x > 0 f(x) is positive.
At x = 0, f(x) is not defined. (The function definition does not
define f(0), we have to assume f(x) is not defined at x = 0.)
and
Since these one
sided limits are equal it follows that the limit of f(x) as x
0 is 0.
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12)
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As we saw in example 3,
does
not exist. We
also
saw in example 1 that
exists
and equals 0.
The limits must exist for each term in the limit expression for the
limit of that expressionto exist.
In this case the limit only exists for the first term, x,
but not for the second term 1/x. Therefore, the limit
does not exist for the sum of these terms.
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13)
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This example is the same as example 12 except the limit has x
approaching 1.
Example 1 tells us that the limit of the first term is 1 and
the limit of the second term is also 1. Therefore the limit
exists and equals 1 + 1 = 2.
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We've discussed examples where the limit of a function can increase or decrease without bound.
We've also seen that there can be jumps/breaks in the function where the on-sided limits differ at the break.
In both cases we said that the limit does not exist. Even though the
two cases are different, and we should distiguish them, we usually don't, caring only whether or not the limit
exists, and if so, what that limit is. These cases will become important, however, when we
later discuss derivatives.
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