A Crash Course on Limits (in class worksheet)

Transcription

1 A Crash Course on Limits (in class worksheet) Many its may be found easily by simple substitution. For example: x x x ( ) = f() = and x = f () = 8 7 and x x = f () = So the first rule is to always TRY substitution first. If this gives you a number, you have found the it! ( ). Now try these exercises as practice:. x x x x x RULE : Always try substitution first. x x x x. However, many its cannot be found by simple substitution. These are the ones that we should concentrate on, because they require a little more thought and work. We will look at some of these by categories: I. Finding its when substitution gives. A. Trying substitution to find x x = f () = obviously does not give a correct answer, since is an undefined term and CANNOT be given a value. But we may notice that the numerator can be factored. When we factor it, the fraction can be reduced as follows: x x = (x )(x ) x x last step once again involved doing a simple substitution. = x (x ) = which is the correct answer. Notice that the Now try these exercises as practice for this type of problem:. x x 6 x 9 9 x. x ( ) 6. x RULE :If substitution gives /, use algebraic methods to re-write the function. II. Finding its when substitution gives c. If substitution gives c (and note that this is different from /!), you must look at the it from both the right and left hand sides. Study the following example: x. Note that substitution gives. So we will look to see what happens to the value of the function if we let x approach from both sides. right side, while from the left side: x = - x = = is from the =. Since the value of the function is

2 approaching two different values from the two sides, we may conclude that this it does not exist. Another example: Find ( ) x. This time, substitution gives /. So we again need to look at the its from both sides as follows: From the right side x while from the left side: x x ( ) = ( x ) = =, =. In this one, the it from both sides appear to be approaching positive infinity. The it still does not exist but we can indicate that the values of this function are approaching Positive infinity. Now try these exercises as practice for this type of problem: 7. x x x 8. x x 9. x x. x x RULE :If direct substitution gives c/, you must look at the it from both sides. III. Limits of compound functions When trying to find the it at a point other than where the domain breaks, substitution usually works. But if you must find the it at the point where the function breaks, you must look at the it from both sides., if x > Example: = x, if x < To find the it at x =, substitution works, ie, f() =. To find the it at x =, substitution works, ie, f() =. But to find the it at x =, you must look at the it from both sides - as follows: f(x) = (x ) = f () = while x f(x) = x ( x ) = f () =. Now since the right and left hand its are not the same, we must conclude that the it at x = does not exist. Now try these exercises for practice:. If = x x, if x >, if x <, find these its: a) b) f(x) c) x x f(x) d) x f(x) e) x f) x RULE : To find the it of a compound function at the point where the function "breaks", you will need to look at the it from both sides.

3 IV. Finding its from graphs When trying to find its from the graph of a function, the it is simply the value of the function except at the endpoints of an interval or anywhere there is a discontinuity on the graph. For example, the it of the above function at x = is because f() =, and the it of the function at x = is because f() is. However, it is not quite so simple at x = or - (the endpoints) or at x = - or x = (points of discontinuity). At these points, we must look at the its from both sides. At x = -, for example, the right hand it is - because the value of the function is approaching - as x -> -. However, from the left side, the function is undefined, so the it does not exist. Since the right and left hand its are not the same, we would say that the it at x = - does not exist. At x = -, the left hand it is, but the right hand it is. So the it at x = - does not exist. At x =, the left hand it is, but the right hand it is. Again, the it at x = does not exist. At x =, the left hand it is, but the right hand it does not exist. So the it at x = does not exist. Now, tell what the following its would be, referring to the graph below. Give a reason for each answer.

4 a) x f(x) b) x f(x) c) x f(x) d) x e) x f) g) h) x Rule : When reading its from a graph, look at the it from both sides at either an endpoint or a point of discontinuity. Anywhere else, the value of the function at the point is the it. V. Finding its when x -> infinity A. Basic Rules: a) c = c b) x n c = c) c = x n Examples: a) x = x b) x = c) x x = B. To find g( x), we will use the idea of dominant terms. Select the dominant term in the numerator and the dominant term in the denominator. Then replace f(x) and g(x) with these dominant terms, reduce the fraction to lowest terms, and then find the it of the result, using the basic rules above. Examples: x x a) = x x x x = x =

5 x x x b) = x x x x = x x = x x c) x x x = x x x = x x = After you have studied these examples and have worked a few exercises, you may discover a shortcut! Now try these exercises: x x x x. x x. x x. x x x x RULE 6: To find the it of a rational function as x -> positive infinity or negative infinity, replace each part of the function with its dominant term, reduce completely, and then take the it. Finally, find these its, using any of the rules above: x x 6. x x 8. x x x x x x 7. x x 9. x x x x. f(x) if = x x, if x x x, if x <