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Limits that Fail to Exist

Limits that fail to exist for one of four reasons :
1) One-sided limits are the same as normal limits, we just restrict x so that it approaches from just one side only.Different right and left behavior.
$x->a^{-}$ means x approaches to a from left side.(Left hand limit LHL)
$x->a^{+}$ means x approaches to a from right side.(right hand limit RHL)
2) The given function does not approach to a finite value which is unbounded behavior of the given function. In such limit, f(x) increases or decreases without bound as x approaches to any value 'c'.
3) Some times the function does not approach to a particular value.f(x) oscillates between two fixed values as x approaches to any value 'c'. (Oscillating behavior)
4) The x-value is approaching the end point of a closed interval.

Examples on Limits that fail to exist

Example on different right and left behavior :
$\lim_{x->0}\frac{|x|}{x}$

Solution : According to the definition of absolute value function
|x| = x, x>= 0
      = -x, x < 0

$\frac{|x|}{x}$ = 1 , x > 0

       = -1, x < 0

So no matter how close x gets to 0, there will be positive and negative x values that gives you f(x) = 1 or f(x) = -1.

$\lim_{x->0^{-}}\frac{|x|}{x}$ = -1

$\lim_{x->0^{+}}\frac{|x|}{x}$ = +1

As the value of left side limit and right side limit is different so the limit of the given function fail to exist.Below is the graph of the given function.
Example on unbounded behavior:
$\lim_{x->0}\frac{1}{x^{2}}$
Solution : When we graph the given function we can see that as x approaches 0 either the right or left side , f(x) increases without bound. This means that when we choose the value of x closes to 0, we will get f(x) to as large as we want.So here the limit fail to exist. Example on oscillating behavior :

$\lim_{x->0}sin\frac{1}{x}$
Solution : Let f(x) = $sin\frac{1}{x}$ we can see that as x approaches to 0 the the sine value oscillates (fluctuates) between -1 to 1 so the limit does not exist because no matter how small you choose the value of x.


12th grade math

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