Algebra of Limits and Algebraic Limits

Transcript

0:00Introduction

Hello everyone! First, a small puzzle. Put x equal to two in x cubed minus eight, over x squared minus four. The top is eight minus eight, zero, and the bottom is four minus four, also zero. So is the limit as x tends to two zero? One? Or does it not exist? Hmm, every guess is risky. By the end of this video, you will find the exact value in two lines. Limits like this come up often in JEE, so let's begin.

0:34Algebra of limits

So, first the rules. Lim is short for limit, and the arrow below it means tends to. Say f of x tends to capital L, and g of x tends to capital M, as x tends to a. Notice the condition: both limits are finite. With me? Then limits behave nicely. The limit of f plus or minus g is L plus or minus M; that sign, a plus above a minus, means plus or minus. And the limit of f times g is L times M, okay? Now the fraction, f over g. Can we always say that its limit is L over M? Only when M is not zero! The slashed equals sign means not equal to. If M is zero, this rule says nothing, and we must work harder. And powers: the limit of f to the power n is L to the power n. Clear? One more. If h is continuous at capital L, the limit of h of f of x is h of capital L: the limit goes inside h. And remember, all these rules need finite limits first. Right?

1:46Indeterminate forms: rewrite first

Now, when these rules fail, we land on forms like these. Zero over zero, infinity over infinity, infinity minus infinity, zero times infinity; the sideways eight is infinity, and the dot means times. Then the one to the power infinity form, zero to the power zero, and infinity to the power zero. These are the indeterminate forms. Got it? So, tell me honestly. If putting x equal to a gives zero over zero, is the limit zero? No, no, careful! Indeterminate means the form alone does not decide the value; these forms need rewriting first. Writing zero the moment you see zero over zero is a common trap. And one to the power infinity is not simply one either. With me?

2:39Zero over zero: factorise and cancel

So how do we rewrite? Suppose a polynomial becomes zero when you put x equal to a. What does that tell you about the polynomial? Yes, exactly! x minus a is a factor; that is the factor theorem. So if a polynomial top and a polynomial bottom both vanish at x equals a, both have this factor. The double arrow means implies. Okay? So the method is really simple. Factorise, cancel x minus a, then put x equal to a. Cancelling is allowed: x comes close to a but never equals a, so x minus a is not zero. Clear?

3:19Standard result: (x^n - a^n)/(x - a) tends to n a^(n-1)

Now, a result to know by heart. The limit as x tends to a, of x to the power n minus a to the power n, over x minus a, is n times a to the power n minus one. Conditions: n rational, and a greater than zero. Got it? Quick test, from the course. The limit as x tends to two, of x to the power five minus thirty two, over x minus two. Hint: thirty two is two to the power five. Here n is five, and a is two. So the limit is five times two to the power four: five times sixteen, eighty. One line, and done! Perfect! And one more standard result. The limit as x tends to zero, of one plus x to the power n, minus one, over x, is n. Same idea, with one plus x tending to one. Okay?

4:14Surds: multiply by the conjugate

Now, what if the top has square roots, called surds? Take root p minus root q; the tick-shaped sign is the square root. Its conjugate is root p plus root q: the middle sign flipped. Right? Now multiply the two. Root p minus root q, times root p plus root q. What do you get? Correct, it is p minus q! It is the difference of squares, so the roots disappear completely. That is the whole trick. So, multiply the numerator and the denominator by the conjugate. This is rationalisation. Always both, never only the top, or the value changes. Clear?

4:58Example 1: limit of (x^3 - 8)/(x^2 - 4) as x tends to 2

Okay, Example one, an easy one from the course: our puzzle from the start! Find the limit as x tends to two, of x cubed minus eight, over x squared minus four. Pause the video, and try it first. Done? Okay, let's check. At x equals two: eight minus eight, over four minus four, zero over zero. So x minus two is a factor of both. Right? Now factorise. x cubed minus eight is x minus two, times x squared plus two x plus four. x squared minus four is x minus two, times x plus two. Cancel x minus two. We are left with x squared plus two x plus four, over x plus two. Now put x equal to two. Four plus four plus four is twelve, over two plus two, four. Twelve over four: three! Puzzle solved: not zero, not one, but three. See? Now, a topper's trick. Eight is two cubed, and four is two squared. So the top and the bottom both look like our standard result. Can it give the answer faster? Yes! Divide the top and the bottom by x minus two; the sign with a dot above and below means divide. The top tends to three times two squared, twelve; the bottom to two times two, four. Three again! And the quotient rule is fine, since four is not zero.

6:30Example 2: rationalising a surd limit as x tends to a

Got it? Now Example two, a little harder, also from the course. Find the limit as x tends to a, of root of three x minus a, minus root of x plus a, all over x minus a, with a greater than zero. Pause the video, and try it. Done? Let's check. At x equals a, the top is root two a minus root two a, zero, and the bottom is a minus a, zero. Zero over zero again. With me? Surds on the top, so we rationalise. Multiply above and below by the conjugate: root of three x minus a, plus root of x plus a, okay? Now the top becomes three x minus a, minus the bracket x plus a. Many students write two x here. Is that right? Oh no, careful! The minus sign goes to both terms: minus x, and minus a. So the top is two x minus two a, which is two times x minus a. There is our factor. Clear? On the bottom, x minus a times the sum of the two roots. Cancel x minus a, since x is not equal to a. We are left with two, over the sum of the two roots. Right? Now put x equal to a. Each root becomes root two a, so we get two over two root two a: one over root two a. And as a is positive, this is a real number. Perfect!

8:03Quick revision: algebra of limits, 0/0 form, key results

So, quick revision. The algebra of limits needs finite limits, and a non-zero bottom limit for a quotient. Zero over zero is not an answer: factorise and cancel, or rationalise. And the standard results: x to the power n minus a to the power n, over x minus a, tends to n times a to the power n minus one; and one plus x to the power n, minus one, over x, tends to n. Now go and try the practice problems on this page. All the best!