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# Functions continuous at specific x-values

AP.CALC:
LIM‑2 (EU)
,
LIM‑2.B (LO)
,
LIM‑2.B.1 (EK)
,
LIM‑2.B.2 (EK)

## Video transcript

which of the following functions are continuous at x equals 3 well as we said in the previous video in the previous example in order to be continuous at a point you at least have to be defined at that point we saw our definition of continuity f is continuous at a if and only if the limit of F as X approaches a is equal to f of a so over here in this case we could say that a function is continuous at x equals 3 so f is continuous at x equals 3 if and only if if and only if the limit as X approaches 3 of f of X is equal to f of 3 now let's look at this first function right over here natural log of X minus 3 well try to evaluate it's not f now it's G try to evaluate G of 3 G of 3 right here G of 3 is equal to the natural log of 0 3 minus 3 this is not defined you can't raise e to any power to get to 0 you try to go to you know you could say negative infinity but that's not that's not defined and so if this isn't even defined at x equals 3 there's no way that it's going to be continuous at x equals 3 so we could rule this one out now f of X is equal to e to the X minus 3 well this is just a shifted over version of e to the X this is defined for all real numbers and as we saw in the previous example it's reasonable say it's continuous for all real numbers and you could even do this little test here the limit of e to the X minus 3 as X approaches 3 well that is going to be that is going to be e to the 3 minus 3 or e to the 0 or 1 and so f is the only one that is continuous and once again it's good to think about what's going on here visually if if you like both of these are you could think of it this is a shifted over version of Ln of X this is a shifted over version of of e to the X and so if we like we could draw ourselves so I'm axes so that's our y-axis this is is our x-axis and actually let me draw some points here so that's one that is one that is let's see two three two and three and let's see these are I said these are shifted over aversions so actually this is maybe not the best way to draw it so let me draw it this is one two three four five and six and on this axis I won't make them with the same scale let's say this is one two three I'm going to draw one two three I'm gonna draw a dotted line right over here so G of X Ln of X minus three is going to look something like this if you put three in it it's not defined if you put four in it Ln of four well that's our Ln of four minus one so that's going to be Ln of Ln of four minus three is actually me just draw a table here I know I'm confusing you so if I say X and I say G of X so at three you're undefined at four this is Ln of one Ln of 1 which is equal to zero so it's right over there so G of X is going to look something like something like that and so you can see at three its you have this discontinuity there it's not even defined to the left of three now f of X is a little bit more straightforward if you have so e to the three is going to be or sorry e F of 3 is going to be e to the 3 minus 3 or e to the zero so it's going to be one so it's going to be looked something like this it's going to look something something like like that there's no jumps there's no gaps it is going to be continuous that frankly all real numbers so for sure it's going to be continuous at 3