Limits of composite functions - How to find them?

In this video we’re talking about how to find the limit of composite functions.

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0:33 // Two different ways to write a composite
1:01 // Two ways to find the limit of a composite: algebraically and graphically
1:28 // How to find the limit of a composite algebraically
4:10 // How to find the limit of a composite when you have the graphs
6:52 // Composites can be given as one function or as two functions (together vs. separately). How to split a composite function apart? How to create a composite function from two functions?
9:33 // Limit laws still apply when you’re finding the limit of a composite function
11:46 // An example with a rational function
15:46 // What to do when the limit doesn’t exist (DNE). What do you do when the one-sided limits are different? What if the inner function of the composite doesn’t have a general limit?
19:22 // What do do when the outer function of the composite doesn’t have a general limit?
20:44 // What to do when the graph doesn’t have a general limit, or the one-sided limits are different?

Remember that composite functions are “functions of functions”, which means that we have one function plugged into another function. As an example, sin(x^2) is a composite function because we’ve plugged the function x^2 into the function sin(x). Think of any function that as an “outer part” and an “inner part” as composite functions.

The most important thing to know about evaluating limits of composite functions is that we can do so algebraically or graphically. Basically that means we can do math to figure out the limit, or we can look at the graphs of the two functions (if we have them) and figure it out that way. In this video we’ll look at examples of both so that you know how to find the limit using math, and also how to find the limit just by looking at the graphs.

In either case, whether you’re looking at the limit algebraically or graphically, you’ll be finding the limit of the inside function, and then evaluating the outside function at the resulting value. So if you’re taking the limit as x goes to 0 of the composite f(g(x)), that means you’ll be finding the limit as x goes to 0 of g(x), and then taking whatever real-number result you got and plugging that number into f(x). Whatever you get will be the answer to the problem.

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Hi, I’m Krista! I make math courses to keep you from banging your head against the wall. ;)

Math class was always so frustrating for me. I’d go to a class, spend hours on homework, and three days later have an “Ah-ha!” moment about how the problems worked that could have slashed my homework time in half. I’d think, “WHY didn’t my teacher just tell me this in the first place?!”

[Dave]

Hi Krista, how do you do all this good work. I'm desperately trying to do engineering maths whilst taking medication for mental illness. I try so hard to understand the math yet I often feel like I am repetitively hitting a brick wall

Keep up the good work Krista your awesome

[georgepolasky]

You are most welcome, Krista. I look forward to your videos and enjoy viewing them all. I do so admire you, and you challenge my brain, and you have such a sweet personality.

[christopherramsey]

The subject was so clearly explained ! Very helpful ! Thank You !

[kimjin]

Great review before the summer break ends

[bayarbotany]

Unbelievably amazing and simple explanation!, Thank you very much

[georgepolasky]

So brilliant and also so very nice at the same time. A combination not found often.

[angusriffraff]

Early in the video I think you said that you can move the limit to the inner function provided that the outer function is continuous at the limit of the inner function. The limit of xsinx as x goes to 0+ was found to be 0. Ln, the outer function, isn't continous at 0. Am I missing something (like maybe it's okay because you're only taking the right sided limit)? Thanks in advance for a reply as that part has me confused.

[tb]

Confused about the first one, lim as x approaches 0 from the right of ln(xsinx). Wouldn't it be undefined/DNE since ln(0) = ln(lim xsinx as x approaches 0) = undefined? Aren't we using limit properties to only find the limit of xsinx as x approaches 0?

[anthonyzurita]

Thank you. The question I had to resolve was g(f(x)) but this helped me understand the concept and allowed me to answer my hw.

[boboganbobogan]

when we take the limit of the inner function and then we plug in the value of the inner function as x approaches some value in the outside function and we get some value. Is this value the point of x where outside function is defined (is this what we want to find?) or do we want to find what value is x approaching of the outside function when the inner function approaches some value?

[jorgei.alonso]

I would like to comment on the continuity condition: the reason why it's there is because it is possible to have both a limit for g at p and a limit for f at q but with f(g(x)) having a different limit at p. For example, let g(x) = 5 for all x, f(x) = 4 if x is not 5 and f(5)=5. Then the limit of g at 2 is 5, the limit of f at 5 is 4 but the limit of f(g(x)) at 2 is equal to the limit of f at 5 which is 5.

[ankitbisht]

Help me in modern algebra please explain group theory i beg you ((((

[marbangens]

my calculator say this is actually minus infinity. and also for clarity I think its better to write, ln(x*sin(x))... but it is the same thing... but minus infinity is a very negative answer :) but I got it. thanks for you're video :)

[muchacamara]

*You look so beautiful, Krista. Thank you for uploading great content.*

[xiangli]

2:01 I don't understand why you can do that. Please provide a proof.

[marbangens]

hahaha e to the power of dose not exist. this math is funny :)

[legendaryxk]

what if they show appocalyptic survival world bit more like kingdom of the planet of the apes

[br]

You did not discuss some hardest examples. This is very elementary.