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Graphs of the reciprocal function (solid curve) and its negative (dashed curve) in the first and fourth quadrants of the Cartesian plane, where x > 0 Each axis division represents 1 unit
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and f (x) = x 1 As the value of x increases without end, both of these functions approach, but never reach, 0 We can therefore write Lim f (x) = 0
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x
and Lim f (x) = 0
x
As the value of x becomes arbitrarily small but remains positive, both of these functions approach singularity The reciprocal function blows up positively while the negative reciprocal function blows up negatively Therefore, we must conclude that neither Lim f (x)
x 0
nor Lim f (x)
x 0
exists It s tempting to claim that Lim f (x) = +
x 0
and Lim f (x) =
x 0
However, we haven t explicitly defined + ( positive infinity ) or ( negative infinity ), so such statements are informal at best
Right-hand limit at a point Consider again the reciprocal function
f (x ) = x 1 To specify that we approach 0 from the positive direction, we can refine the limit notation by placing a plus sign after the 0, as follows:
x 0+
Lim f (x )
This expression reads, The limit of f (x ) as x approaches 0 from the positive direction We can also say, The limit of f (x ) as x approaches 0 from the right (In most graphs where x is on the horizontal axis, the value of x becomes more positive as we move toward the right)
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This sort of limit is called a right-hand limit Because f is singular where x = 0, this particular limit is not defined
Left-hand limit at a point Let s expand the domain of f to the entire set of reals except 0, for which f is not defined because 0 1 is meaningless Suppose that we start out with negative real values of x and approach 0 from the left As we do this, f decreases endlessly Another way of saying this is that f increases negatively without limit, or that it blows up negatively Therefore,
x 0
Lim f (x )
is not defined We read the above symbolic expression as, The limit of f (x) as x approaches 0 from the negative direction We can also say, The limit of f (x) as x approaches 0 from the left This sort of limit is called a left-hand limit
An example Let s consider a function g that takes the reciprocal of twice the independent variable If the independent variable is x, then we have
g (x ) = (2x ) 1 Imagine that we allow x to be any positive real number As x gets arbitrarily large positively, g (x) gets arbitrarily small positively, approaching 0 but never quite getting there We can say, The limit of g (x ), as x approaches infinity, is 0, and write Lim g (x) = 0
x
This scenario is similar to what happens with the reciprocal function, except that this function g approaches 0 at a different rate than the reciprocal function as the independent variable becomes arbitrarily large Now let s see what happens when x gets smaller but stays positive, so that g (x) gets larger If we make x close enough to 0, we can make g (x) as large as we want This function, like the reciprocal function, blows up as x approaches 0 from the positive direction, but at a different rate Therefore
x 0+
Lim g (x )
is not defined
Another example Suppose that x is a positive real-number variable, and we want to evaluate
Lim 1/x 2
x
Let s start out with x at some positive real number for which the function is defined As we increase the value of x, the value of 1/x 2 decreases, but it always remains positive If we choose some tiny positive real number r, no matter how close to 0 it might be, we can always find
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