00:01
Hi, in this question we have this function f of x, which is defined to be 1 over 1 minus the absolute value of x.
00:12
And we want to find points where this function is not defined and therefore not continuous, and then for these points or point, say whether the discontinuity is removable or not.
00:25
So it's clear that f of x is not defined when the denominator is zero.
00:36
Not defined when 1 minus the absolute value of x is 0.
00:47
But this happens only, of course, when the absolute value of x is 1, which happens if and only if x is plus or minus 1.
01:04
And then to determine whether the discontinuity is removable, we have to consider the limits.
01:10
So the limits as x approaches negative 1 from the left, maybe i'll just write the function minus the absolute value of x.
01:24
So when we're approaching negative 1 from the left, left then our values are like negative 1 .01 or something like that in which case the absolute value is positive 1 .01 and so in either case this is going to be 1 minus so in this case this is going to be greater than 1 and therefore 1 minus it will be negative so this limit is going to be negative infinity and then as we approach negative 1 from the right now if we're approaching from right then we're looking at values like negative 0 .99 and so the absolute value is going to be like 0 .99 so these values are going to be less than 1 so the denominator will be positive and so this is going to be positive infinity and therefore x equals negative 1 is not removable it's not a removable discontinuity and then for x equals 1 we do the same thing and it's basically the same idea so but now as we approach one from the left, the absolute value of x is going to be less than one...