​Let's say we ask you to find a limit at a number...

• ​your first line of defense would be to try and substitute the x-value into the function then...

  • ​if you get a number; that is the limit

  • ​if you get "undefined", you should try to get the limit by manipulating the function algebraically or by graphing and analyzing the graph

​The correct answer is that at -4 the function is undefined, so we need to manipulate it to get the limit

​Let's factor the denominator and see what happens

​Factoring the denominator

​

​Let's try another one...

​If you were to try step one, and substitute x = 1, you would get 0 in the denominator and the function would be undefined.

​

​

​

​

​

​

​Instead, we need to manipulate the numerator to hopefully eliminate x-1 from the denominator.

​Now that we have a new numerator...

​Now that the fraction is fully simplified, we can analyze the limit. In this case, the limit at 1 is 1, because that is what the