Essential Knowledge 1.1C1
Essential Knowledge 1.1C1 Students will know that limits of sums, differences, products, quotients, and composite functions can be found using the basic theorems of limits and algebraic rules
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Honestly I wouldn't bother thinking so hard about these, but here are some official rules if you want to be mathematically perfect about your answers.
Foundational Rules (I'm making up the names of these)
Constant Rule
Some random number doesn't care if it's part of a limit.
Example
why would the 7 care what x is doing if it's not interacting with x?
Identity Rule
Once you've got x by itself, you can plug in the value it's approaching. For me, this is a bit overkill. I usually just plug in c for all my x's right away and see what happens.
Example
Trigonometry Rule, Logarithm Rule, Exponent Rule
If you have a trigonometry function, a logarithmic function, or an exponential function, limits can slide right in.
Examples
As seen in the foundational rules, x has to be isolated before you can take the limit. If we're going to solve limits the official way, we'll need to learn a few ways to break large problems into smaller pieces. But before you can accomplish any of this, you'll need to understand the basics from 1.1CP
There are some rules to help us do this. They go by many names. When I was in calculus, my teacher called them the "Limit Laws". College Board refers to them as "Theorems of Limits". Your textbook calls them "Properties of Limits". I've also seen "Limit Rules". In the end, it's so simple who really cares.
Basically these rules are just the distributive property. You can take a limit with lots of parts and turn it into a bunch of smaller limits. That's all I think when I do these and it never fails me.
Limit Laws
Basically, just distribute. Larger problems require more rules, one after another.
For all of these laws, and
Constant Multiple Rule
Remember that k doesn't care. Just distribute the limit.
Example
(used Constant Multiple Rule)
(used Identity Rule)
= 8
Sum Rule
In other words, the limit of the sum of two functions = the sum of the limit of each function seperately. Just distribute the limit.
Example
(used Sum Rule)
(used Constant Multiple Rule)
(used Identity Rule)
= 18
Difference Rule
In other words, the limit of the difference of two functions = the difference of the limit of each function seperately. Just distribute the limit.
(used Difference Rule)
(used Sum Rule)
(used Constant Multiple Rule)
(used Identity Rule)
= 6
Product Rule
In other words, the limit of the product of two functions = the product of the limit of each function seperately. Just distribute the limit.
Example
(used Difference Rule)
(used Sum Rule)
(used Product Rule)
(Trigonometry Rule)
(Constant Multiple Rule)
= (4)(2) + (5)(2) - (6)(2)(sin(2π)) (Identity Rule)
= 8 + 10 + (12)(0) = 18
Quotient Rule
In other words, the limit of the quotient of two functions = the quotient of the limit of each function seperately. Just distribute the limit.
Example
(used Sum Rule)
(used Quotient Rule)
(used Constant Rule)
(used Constant Multiple Rule)
(used Identity Rule)
(used stuff from EK1.1CP)
(used something you haven't learned yet called L'Hospital's Rule. No, its not just cancelling infinities that doesn't always work)
= 0.5
Power Rule
In other words, the limit of the term with an exponent= the limit of of the base, raised to the exponent. Just put the limit inside.
Example
(used Quotient Rule)
(used Constant Rule)
(used Constant Multiple Rule)
(used Power Rule)
(used Identity Rule)
