L IMITS EXPLAINED S TUDYING THE BEHAVIOR OF FUNCTIONS

T HERE ARE BASICALLY TWO TYPES OF LIMITS THAT GO INTO FINDING A LIMIT. A left hand limit is denoted like this : with the little uppercase negative behind a. A right hand limit is denoted like this : with the little uppercase positive behind a. Both of these must match and be the same number in order to find: (Notice there is no “+” or “-“ behind the “ a”.) If the left hand limit and the right hand limit do not match and are not the same number then does not exist.

is the y-value on the y-axis that the function gets close to as x gets close to a from the left hand side of a on the x axis. In other words, as approaches x approaches a from the left, the function approaches some y-value from the left. (Use your finger to follow what the function is doing on the left hand side of a = 2 in the picture) Consider. This would mean we want to find the y – value that the function gets close to as x approaches 2 from the left hand side of 2 on the x axis. (Use your finger to follow the function as the value of x get closer to x = 2 on the x axis.) That would be y = 1. So, = 1.

is the y-value on the y-axis that the function gets close to as x gets close to a from the right hand side of a. In other words, as x approaches a from the right, the function approaches some y-value from the right. (Again, use your fingers to follow the function). So, consider. Follow with your finger the function as you let x approach 2 from the right side of 2, and you approach the y – value y = 1 also. Therefore, = 1 as well.

So for the graph below, as you approach x = 2 from the left or from the right side, the function value (y – values) approaches y = 1. Therefore, we then can say that the limit as x approaches 2 regardless of direction, denoted is: = 1.

P OINTS TO REMEMBER If, =, then = that y- value!!! -AND- If,, then does not exist!!! Let’s look at an example of this second possibility. We will use the same graph as before.

Using this same function, let’s find the following: = 2 since the y – value that the function approaches as x approaches 0 from the left is y = 2. = -1 since the y – value that the functions approaches as x approaches 0 from the rights is y = -1. Since the left hand limit and the right hand limit are different, we say that: does not exist.

T O EVALUATE LIMITS WITHOUT DRAWING PICTURES, HERE ARE THE STEPS YOU TAKE : Step 1 : Simply plug the value of x your approaching into the place of x. Do the arithmetic. If you get a defined number, even a decimal or fraction, then that number is the limit value and you are finished. Example :, so 7 is the answer. Example:, so 10 is the answer.

Step 2 : If you get, when you plug in the value of x, you must do some algebra. Try factoring and canceling, rationalizing and canceling, or simplifying and canceling. After canceling, plug the value of x in again and you should get a number. That number will be the limit. Example :, so 2 is the answer. Notice in step 2 that I cancelled out ( x – 1). Also, notice that the “limit” symbol was carried from step to step until I was able to actually plug in the x = 1 and get a value. Example: = = = So, ¼ is the answer. Notice in step 2 the top portion was “FOILed” but not the bottom. I then canceled out (x – 4), in step 3. Again the “limit” symbol was carried from step to step until I was able to actually plug in the x = 4 and get a value.

Step 3 : If you get, you must either draw the picture in your graphing calculator, if you have one, or do what I like to call PLUG-N-CHUG. Example: To find gives you when you plug in x = 2. So use your calculator to plug in numbers that get close to 2 coming from the left. That would be x = 1, x = 1.5, x = 1.9, x = 1.99, etc. (These numbers steadily get closer to 2 from the left). Plug them each into the place of x in using a calculator and you will get these numbers: -5,-10, -50,-500. As you can tell the numbers are getting more and more negative so = To find you will get close to 2 from the right with x = 3, x = 2.5, x = 2.1, x = Plugging these numbers into the fraction gives 5, +10, +50,+500, so =

Since ∞ ≠ - ∞, we know that does not exist!! (notice there is no “+” or “-“ behind 2).