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qr code vb.net open source f (x) = x 1 in .NET framework
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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 Righthand 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) Sequences, Series, and Limits
This sort of limit is called a righthand limit Because f is singular where x = 0, this particular limit is not defined Lefthand 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 lefthand 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 realnumber 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

