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## Determining limits using algebraic properties of limits: direct substitution

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# Limits of piecewise functions

AP.CALC:

LIM‑1 (EU)

, LIM‑1.D (LO)

, LIM‑1.D.1 (EK)

## Video transcript

- [Instructor] Let's
think a little bit about limits of piecewise functions
that are defined algebraically like our f of x right over here. Pause this video and see
if you can figure out what these various limits would be, some of them are one-sided, and some of them are regular
limits, or two-sided limits. Alright, let's start with this first one, the limit as x approaches four, from values larger than equaling four, so that's what that plus tells us. And so when x is greater than four, our f of x is equal to square root of x. So as we are approaching
four from the right, we are really thinking about
this part of the function. And so this is going to be
equal to the square root of four, even though right at four, our f of x is equal to this, we are approaching from
values greater than four, we're approaching from
the right, so we would use this part of our function definition, and so this is going to be equal to two. Now what about our limit of f of x, as we approach four from the left? Well then we would use this
part of our function definition. And so this is going to
be equal to four plus two over four minus one, which is equal to 6 over three, which is equal to two. And so if we wanna say
what is the limit of f of x as x approaches four, well
this is a good scenario here because from both the left and the right as we approach x equals
four, we're approaching the same value, and we
know, that in order for the two side limit to have
a limit, you have to be approaching the same thing
from the right and the left. And we are, and so this is
going to be equal to two. Now what's the limit as x
approaches two of f of x? Well, as x approaches
two, we are going to be completely in this
scenario right over here. Now interesting things do
happen at x equals one here, our denominator goes to
zero, but at x equals two, this part of the curve
is gonna be continuous so we can just substitute
the value, it's going to be two plus two, over two minus
one, which is four over one, which is equal to four. Let's do another example. So we have another piecewise function, and so let's pause our video
and figure out these things. Alright, now let's do this together. So what's the limit as x
approaches negative one from the right? So if we're approaching from the right, when we are greater than
or equal to negative one, we are in this part of
our piecewise function, and so we would say, this
is going to approach, this is gonna be two, to
the negative one power, which is equal to one half. What about if we're
approaching from the left? Well, if we're approaching from the left, we're in this scenario right over here, we're to the left of
x equals negative one, and so this is going to
be equal to the sine, 'cause we're in this case,
for our piecewise function, of negative one plus one,
which is the sine of zero, which is equal to zero. Now what's the two-sided
limit as x approaches negative one of g of x? Well we're approaching
two different values as we approach from the right, and as we approach from the left. And if our one-sided
limits aren't approaching the same value, well then
this limit does not exist. Does not exist. And what's the limit of g of x, as x approaches zero from the right? Well, if we're talking
about approaching zero from the right, we are
going to be in this case right over here, zero is
definitely in this interval, and over this interval,
this right over here is going to be continuous,
and so we can just substitute x equals zero there, so it's
gonna be two to the zero, which is, indeed, equal
to one, and we're done.

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