Finding limits

Finding limits

by Karine Ptak

​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

Multiple Choice

For example:  find the limit of the function in the picture by substituting 2 for x.  What y-value did you get for your limit?

• undefined

• 0

• 4

• -2

Multiple Choice

For example:  find the limit of the function in the picture by substituting -4 for x.  What y-value did you get for your limit?

• undefined

• 0

• 4

• -2

​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

​

Multiple Choice

Now evaluate the function at x = -4.  What do you get?

• undefined

• -4

• -1/3

• DNE

​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.

Multiple Choice

What is the conjugate of the numerator?

• $$\sqrt[]{x+1}-\sqrt[]{2}$$  

• $$\sqrt[]{x+1}+\sqrt[]{2}$$  

• $$\sqrt[]{x-1}+\sqrt[]{2}$$  

• $$\sqrt[]{x-1}-\sqrt[]{2}$$  

Multiple Choice

What is $$\left(\sqrt[]{x+1}-\sqrt[]{2}\right)\left(\sqrt[]{x+1}+\sqrt[]{2}\right)$$ ? 

• $$\left(x-1\right)+2$$  

• $$\left(x-1\right)-2$$  

• $$\left(x+1\right)+2$$  

• $$\left(x+1\right)-2$$  

​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

Multiple Choice

Try one on your own?  

• 0

• undefined

• $$\frac{1}{2\sqrt[]{2}}$$  

• $$\frac{1}{4}$$