# 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.

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