Overall Limit

Unit 1: Limits and Continuity Limit Notation Overall Limit

Overall Limit Notation:

$\lim_{x \to a} f (x) = y - value$

Read: The limit of $f (x)$ as $x$ approaches $a$ equals $y$ .

The key to recognizing that you are looking at an overall limit will be the lack of an exponent on your given $a - value$ .

Like always, the $a - value$ that you are headed towards is an $x - value$ , and the answer to the limit question is always a $y - value$ .

An overall limit is asking you what $y - value$ does it look like your $f (x)$ is heading towards as you approach the given $a - value$ from both directions, the left direction and the right direction .

What this means is that in order to answer the overall limit question you really need answer the question:

Does the Left-Hand Limit = the Right-Hand Limit?

$\lim_{x \to a^{-}} f (x) = \lim_{x \to a^{+}} f (x) ?$

The answer to this question is either Yes or No.

Option 1: The answer to that question is Yes.

The result of the LHL is the same $y - value$ as the RHL.

Left-Hand Limit = Right-Hand Limit

$\lim_{x \to a^{-}} f (x) = \lim_{x \to a^{+}} f (x) = y - value$

Option 2: The answer to that question is No.

The result of the LHL is not the same $y - value$ as the RHL.

Left-Hand Limit ≠ Right-Hand Limit

$\lim_{x \to a^{-}} f (x) \neq \lim_{x \to a^{+}} f (x)$

Result: The overall limit exists , and it is the same $y - value$ that you found with the left-hand limit and right-hand limit.

Left-Hand Limit = Right-Hand Limit = Overall Limit

$\lim_{x \to a^{-}} f (x) = \lim_{x \to a^{+}} f (x) = y - value$

$\lim_{x \to a} f (x) = same y - value$

Result: The overall limit does not exist. The $y - values$ that you found for the left-hand limit and right-hand limit are different.

Left-Hand Limit ≠ Right-Hand Limit : Overall Limit Does Not Exist

$\lim_{x \to a^{-}} f (x) \neq \lim_{x \to a^{+}} f (x)$

$\lim_{x \to a} f (x) = Does Not Exist (DNE)$

I like to think about an overall limit like my two hands, left and right. If my two hands going to come together, run into each other, at the given $x - value$ , then the overall limit exists and the answer is the $y - value$ where my two hands met. Or my two hands miss each other, don’t run into each other, at the given $x - value$ , then the overall limit does not exist .

Remember, math people are not creative namers. If it is called the left-hand limit it is because that is what your left hand is doing. If it is called the right-hand limit it is because it is what your right hand is doing.

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Unit 1: Limits and Continuity

Overall Limit

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