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How to Find Limits Of PIECEWISE functions
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Feb 20, 2025
During my latest tutoring session, I was working with my student on understanding how to find limits of PIECEWISE functions. We also worked on understanding partial limits and how they affect the continuity of a function If you need help in math, be sure to check out http://tayibs.com/get-math-help/
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now this morning we are working on
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finding limits of piecewise function and
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the most important thing is to
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understand how to define a function over
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the given interval right now this is a
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piecewise function f of X is X square
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minus 6 when X is less than 0 that means
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0 and everything going to work negative
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infinity and then f of X is negative 6
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when X falls between 0 & 5 and then is
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square root of x plus 4 when X is
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greater than or equal to 5 meaning if X
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is 5 for Google was positive infinity
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now you are asked to find a limit of a
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function as X approaches negative
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infinity that means as the function goes
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further and further and further further
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that was negative infinity what is f
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what's the trend of this function of
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what it was happening to his function
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it's gonna keep called towards negative
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infinity when X is ghost was negative
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infinity right so let's say you plug in
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negative 8 thousand years let's let's
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try to if you put in negative 1000 in
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there right
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you'd be negative 1000 squared minus 6
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right negative 1000 squares like a
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million isn't it convenient minus 6 is
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almost like is 909 9 I know something
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something maybe not gonna do all that
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well I mean is positive right so let's
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say you go to negative 1 billion what's
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going to happen to the function it's
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gonna turn into a positive so the
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further you go to towards infinity on a
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negative side the function is going to a
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positive what infinity that's a reason
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why is the saying this is increasing
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without a bond meaning it keeps going
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the further you go from 0 on the
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negative side the because this is
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negative comment this is x squared
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anything negative here will make it
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bigger by positive because any number
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negative squared or raised to an even
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power is always gonna be positive right
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so now you're gonna go to a positive
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infinity the further you go away from 0
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that make sense yeah I think like think
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what you mean is there like
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we're trying to understand why this is
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happening so the function is X square
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minus 6 right when X is less than 0 so
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now as you go closer and closer to
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negative infinity which is on the left
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side going this way we want to choose
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like a random number let's say we put
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negative 100,000 into this function
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what's going to happen to the function
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positive
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well positive what are you G's that
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number if you put negative 100,000 right
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and then you square it it's going to
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give you what a huge amount this is
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gonna give you this number and I'm so
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one more five one zero one two three
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four five right it's gonna give you this
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and this number right this is what you
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get right yeah how big it is number what
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do you know but how huge is number a
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bigger this number oh okay it's like
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positive the billion right now positive
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the billion minus six is always going to
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be positive something close to a billion
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now the further you go from this number
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let's say you go to negative one hundred
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thousand hundred negative a million and
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then you square that you're gonna get
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even a bigger number
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now let's say we go to negative one
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trillion and we square that we're gonna
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get even a huge number so the closer you
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go to negative infinity which is
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infinite because you keep going the
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number is becoming bigger and bigger and
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it's getting close to positive infinity
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that's what it's saying yet okay so
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anything less than zero towards negative
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infinity will give you positive infinity
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okay next thing you're trying to figure
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out is this limit of the function the
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same function this piecewise function as
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X approaches 0 from the negative side
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and we shade we are coming from the left
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side right so now since we are coming
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from the left side we're going to be
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using this function here right now plug
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in 0 there and see what happened
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if you couldn't be negative negative six
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so when X gets closer to 0 from the
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negative side the function gets a little
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good get close to what 8 is negative 6
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this is why you got negative 6 makes it
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through the same function and now we
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have a positive 0 so we are coming from
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the right side correct and then use the
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same piecewise function so what function
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are we gonna use this first one the
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second one or the third one which one
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yeah for that one
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second one right and then what does the
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functions say when X is between 0 & 5
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what's the value of the function
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negative 6 so that's not gonna change
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you see it gonna be negative gonna cave
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in doing it there's nothing you could do
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is a constant function is negative 6 so
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it's continuous it knows no continues a
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constant function between 0 & 5 is
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always going to be negative 6