Function Limits II: When Function Limits Don't Exist
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There are two important scenarios for which a function limit does not exist. The first is when the limit produces two different values dependent upon whether we approach a point from the left or the right. The next picture demonstrates this.
For a function limit to exist it must have the same value if we approach it from either the left or the right. Otherwise, we say that the limit does not exist.
Approaching a point from the left is referred to as a left-hand limit and has the notation
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Notice the minus sign appended to <Sign in to see all the formulas> below $\lim$. In English we would say that "as $x$ approaches $a$ from the left, the function approaches $L_l$." Similarly, approaching a point from the right is referred to as a right-hand limit and has the notation
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Notice the plus sign appended to <Sign in to see all the formulas> below $\lim$. In English we would say that "as $x$ approaches $a$ from the right, the function approaches $L_r$." The limit at point $a$ does not exist if the limit from the left does not equal the limit from the right. The only way the limit can exist is when <Sign in to see all the formulas>.
In the picture above, we see that
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and
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Because the limit from the left and the limit from the right are not the same the limit at point $a$ does not exist.
The second scenario is when a function approaches infinity. See the picture below.
An example of this type of function would be
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where the function gets larger and larger because the denominator gets smaller and smaller as <Sign in to see all the formulas>. In this case we write
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We can have situations where the two scenarios above are combined to have functions like
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where
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and
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Neither the left or right sided limits are equal and even if they were the function is going to infinity! See the graph below.
Remember to keep these pictures in your head. Even though the next lesson will provide a mathematical definition of function limits, it is important to keep the pictures in your head to form some intuition about when a function limit exists.
