WEBVTT
00:00:00.420 --> 00:00:06.550
Find the limit as π₯ tends to zero of π₯ minus four squared minus 16 over π₯.
00:00:07.380 --> 00:00:12.950
The first thing to try here is direct substitution: can we just plug zero in to the expression?
00:00:13.710 --> 00:00:20.350
We replace π₯ by zero and get zero minus four squared minus sixteen all over zero.
00:00:21.330 --> 00:00:26.910
Zero minus four is negative four and negative four squared is 16.
00:00:27.560 --> 00:00:35.790
So simplifying further, we find that itβs just directly substituting zero into the expression, we get the indeterminate form zero over zero.
00:00:36.760 --> 00:00:38.280
Weβre going to have to be more clever here.
00:00:38.680 --> 00:00:42.790
We need to simplify this expression first before substituting in.
00:00:43.690 --> 00:00:45.610
The first thing we do is to distribute.
00:00:46.360 --> 00:00:50.880
π₯ minus four squared becomes π₯ squared minus eight π₯ plus 16.
00:00:51.670 --> 00:00:55.270
We can then cancel the plus 16 and the minus 16.
00:00:56.140 --> 00:00:59.530
So weβre left with just π₯ squared minus eight π₯ in the numerator.
00:01:00.340 --> 00:01:02.860
Both terms in the numerator have a factor of π₯.
00:01:03.030 --> 00:01:08.050
And so we can factor the numerator into π₯ times π₯ minus eight.
00:01:08.830 --> 00:01:16.900
And the factor of π₯ in the numerator cancels with the π₯ in the denominator, leaving us with just π₯ minus eight.
00:01:17.690 --> 00:01:20.780
As these two expressions are equal, their limits must be equal.
00:01:21.460 --> 00:01:32.920
And while we saw thatβs direct substitution on the left-hand side gave the indeterminate form zero over zero, direct substitution on the right-hand side β plugging in π₯ equals zero β gives negative eight.
00:01:33.660 --> 00:01:37.660
And hence, the value of the limit that weβre looking for is negative eight.
00:01:38.560 --> 00:01:49.770
A reasonable question to ask now is if these two expressions are supposed to be equal, then why did plugging in the zero to the left-hand side give a different answer to plugging in zero to the right-hand side?
00:01:50.220 --> 00:01:56.130
On the left-hand side, we have the indeterminate form zero over zero and on the right-hand side, we had negative eight.
00:01:57.050 --> 00:02:09.490
The answer is that in our last step of algebra, where we cancel the factor of π₯ in the numerator with the π₯ in the denominator, we turn something which is undefined when π₯ is zero into something which is defined.
00:02:10.530 --> 00:02:13.910
The two expressions are equal for all nonzero values of π₯.
00:02:14.250 --> 00:02:18.700
But for π₯ is zero, the left-hand side expression is undefined.
00:02:19.900 --> 00:02:25.680
As weβre taking the limit as π₯ tends to zero, we donβt care about what the value of the expression is when π₯ is zero.
00:02:25.980 --> 00:02:28.050
We only care about values of π₯ nearby.
00:02:29.010 --> 00:02:31.450
The limits then are in fact really equal.
00:02:31.640 --> 00:02:35.270
And as weβve seen are in fact equal to negative eight.