Limits

= -As points get closer and closer together, we can see the slopes becoming more accurate. This is an example of **__Limits__**, or **the intended height of a function**. = =-Limits are used to define derivatives, continuity, and integrals.= == = = =-For example:= =//Consider the function f(x).//= =//Because f(x) is differentiable, meaning it has a derivative , and continuous , meaning "you can draw it without lifting your pen," the limit of the function is the value, or height of the function, which is simply f(c).//= = = = = =-However, a function may not always be differentiable.= =Since this function has a sharp curve (a cusp ) at x=CP, the function, g(x), has no derivative. But, fortunately that does not affect the limit.= =Because the function converges at x=CP as it is approached from both sides, the limit is simply the value of CP. So, assuming that CP is the coordinate (3,-1), and= =, .=

=- But, continuity does determine whether a function has a limit or not. = =Since figure 5.2 is not continuous at x=1, there is no limit for x=1. However, there are limits for the value as x=1 is approached from above and below. From below, the intended value (also the real value in this case) for g(x) is 4. Therefore,=

=The same idea applies for the limit from above. Since the intended height of g(x) is 2, the limit of g(x) from above equals 2.= =Therefore,=

= Other times when there aren't any limits: =

=-Limits Do Not occur when the intended height of a function is infinity, or when the function has vertical asymptotes .=

=Let's call the following function f(x). Notice how f(x) has a vertical asymptote at x=-2. There is not a limit for f(-2), f(-2) from above, nor f(-2) from below. From x-values greater than x=-2, the intended height of the function is. Likewise, the limit from below is. So, there is no limit for nor .=

=Now what about the limit of f(-2)? Since the intended height of the value is not the same from above and below, DOES NOT EXIST .=

= Methods to find limits: = = -Substitute the value of the limit in the function. =

=- However, if substituting leads to an indeterminate answer,, you can either try factoring or use L'Hopital's Rule. =

=In L'Hopital's Rule, you derive the numerator and the denominator and reapply the limit.= = = =In factoring, well...you factor!=



= AND REMEMBER... =

= = = Practice! =



=1. What is ?= = A. 3 = = B. 2 = = C. -1 = = D. 2 = = E. The limit does not exist =

= 2. What is ?= = A. 1 = = B. 2 = = C. 3 = = D. 0 = = E. The limit does not exist = = = = 3. What is ?= = A. 2 = = B. 3 = = C. 4 = = D. -2 = = E. The limit does not exist =

= 4. what is ?= = A. 2 = = B. 3 = = C. 4 = = D. -2 = = E. The limit does not exist =

= 5. What is ?= = A. 2 = = B. 3 = = C. 4 = = D. -2 = = E. The limit does not exist =

= ANSWERS =

= 1) B-Although the function's actual value for x=1 is 3, its intended value is 2. And, although it is neither differentiable nor continuous, the function converges from above and below at y=2. = = 2) E-While limits do exist for the function from above and below x=-1, f(-1) does not have a limit because there is a vertical asymptote and the function does not converge. There is no intended height for the function at x=-1 and therefore, the limit does not exist. = = 3) C-When approached from below, the function's intended height at x=1 is 4. Although its actual value happens to be 4, this is not why C is the answer. If the function's actual value of x=1 was 2 instead, the limit for the function from below would still be 4 because that is the end behavior of the function. = = 4) A-Like in #3, the intended height of the function from above is 2. Although the actual value of the function at x=1 is 4, the end behavior of the function from above is 2. = = 5) E-Like in #2, the limit does not exist because there is a vertical asymptote. While the function has limits from above and below due to the intended heights of the function, there is no limit for g(1) because there is no real value. The function does not converge from above and below and the end behaviors of the function do not meet. = = = = Short Answer! =

= 1. Draw a function that is not continuous at f(a), but has a limit of L at f(a). = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Possible Answer: = == = BONUS = = 1. What is ? =

= = = For Answer, scroll down = = = = = = = = = = = = = = = = = = = = = = = = = = = = Answer: All fans of Mean Girls would know that "THE LIMIT DOES NOT EXIST!" = = = THANKS FOR READING