The video discusses topics related to the concept of derivatives and calculus, including the difference quotient and the average rate of change. The formula for the difference quotient is explained, and the concept of limits is introduced through graphs and examples. The video covers one-sided limi...
This video is about the difference quotient and the average rate of change. These are
topics that are related to the concept of derivative and calculus. for function y equals f
of x, a secant line is a lie. that stretches between two points on the graph of the
function. now the slope has the secret line between the two points A, F of a and b, f of
b. a different version represents the average rate of change of a function. the
difference quotient represents the slope of the secant line for the graph of y equals f
of x. it looks like a single entity, but it still represents a difference in x values. let 's work
out a formula for the difference. this video will introduce the idea of limits through some
graphs and examples. in this video, we use the formula f of B. of B minus f of an over
B minus A To calculate an average rate of change. we use the related formula F of X
plus H minus F of X over h to calculate a difference quotient.
The limit as X approaches one of f have X is equal to Ted., but when x is exactly one,
my function is going to have a value of zero and not 10. The limit does not care about
the value of F at one. but the limit does care about what happens for x values on both
sides of a. In general, for any function F of X can guarantee that it lies in an arbitrarily
small interval around l. limits from the left or from the right are also called one sided
limit. as x approaches negative two, our y values are getting arbitrarily large. the limit
as x. goes to two of g of x does not exist because the functions do not approach any
finite number. I prefer to say that these limits do not exist as a finite number, but they
do exist as one. Sided. this video gives some examples of when limits fail to exist. for
this function, f of x graph below, let 's look at the behaviour off of. X in terms of limits
as x approaches negative one, one, and two. negative one and two are the only two
values.
the limit as x goes to zero of sine pi over X or sometimes you 'll see sine one over x. if
you graph this on your graphing calculator and zoom in near x equals zero, you 're
goon an see something that looks roughly like this. it just keeps oscillating up and
down. as x goes towards zero, pi over x is getting bigger and bigger. and that’s wild
behaviour. not a technical term, just to descriptive term. let 's look at an example that
has this wild behavior forcing a limit not to exist. limits as x goes to a off of x and G of
x exist as finite numbers that is not as limits that are infinity or negative infinity. the
limit of the sum is the sum of the limits. and the limit of a quotient is the quotient of
limits. we'll see. In a moment that these conditions hold. the Lemon laws allow us to
evaluate limits of rational functions just by plugging in the number that X is going
towards. as long as that number does n't make the denominator zero, the limit rules
just simply do n't apply. instead, we have to use other techniques to try to evaluate the
limit of a sum or quotient..
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