12. Limits and Derivatives

12 topics3 h

Chapter 12 of NCERT Mathematics for Class 11: Limits and Derivatives. Four sections — Closing in on a value you cannot reach, Computing limits, Derivatives and Two more functions, and their limits. 12 videos, 3 h in all.

Worked answers to the 73 exercise questions in this chapter: Exercise 12.1 · Exercise 12.2 · Miscellaneous Exercise

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Part 1

Closing in on a value you cannot reach

13 minAveraging over shorter and shorter intervals to get a speed at an instantA stone falls for two seconds, and its speed at that exact instant asks for nothing divided by nothing. Every average velocity computed either side of it misses the answer.13 min · 14 parts

14 minApproaching from the left and from the right, and when the two disagreeWalk toward a point on the line from below and the outputs settle on one value; walk in from above and they can settle on another. A limit is a verdict, not a given.14 min · 13 parts

15 minWhy what a function approaches need not be what it equalsx squared minus 4, over x minus 2, has no value at x = 2, and a perfectly definite limit of 4 there all the same. A limit reads what a rule does near a point, never at it.15 min · 14 parts

Part 2

Computing limits

14 minLimits pass through sums, products and quotientsAlmost everything you can compute about a limit rests on one rule: that if you know two limits, you know the limit of the combination. It is not obvious and it is not proved here - so it is scored instead. Nine rules, eighty-one ordered pairs, three operations, and two routes that never see each other's answer: across all 243 scores the number of times they both answered and disagreed is 0.14 min · 12 parts

17 minSubstitution works until it gives nothing over nothing, and then cancellation doesPutting the number in is not the definition of a limit - it is a theorem, and it stops working exactly where nothing over nothing appears. Measured rather than asserted: sixteen rules were read two ways by machinery that never shows one route the other's answer, and substitution matched the run 5 times, quietly gave a different number 2 times, and had nothing to give 5 times more.17 min · 14 parts

16 minTrapping a function between two others to settle the trigonometric casessin x over x, closing in on nothing, is a 0/0 no factorisation can touch - a sine is not built from powers of x. Three areas inside a circle, nested one in the next, settle it instead.16 min · 15 parts

Part 3

Derivatives

17 minThe rate of change at a point, defined as a limit of average ratesSpeed at an instant is a question with no answer as asked - the distance you would divide is nothing and so is the time. The derivative is the repair, and its quotient is 0/0 EVERY time: across 15 functions at 43 places, all 641 readings have a top that shrinks to nothing, and the number substitution can deliver is 0. Measured, not asserted.17 min · 17 parts

19 minRules for differentiating a sum, a product and a quotientDifferentiate x times x by multiplying the derivative of x by itself, and the answer is wrong at thirty-six of thirty-seven tested points, right at one by accident.19 min · 18 parts

18 minThe power rule, and a polynomial's derivative assembled out of it and the sum ruleThe power rule gets proved twice over here, by two arguments sharing nothing but their conclusion - one expands a binomial, the other climbs by the product rule alone.18 min · 18 parts

17 minSine and tangent go back to the definition, because no rule so far reaches themA sine crosses zero infinitely many times, and a ratio of polynomials can only ever cross zero finitely often - so no rule built so far can ever reach it.17 min · 18 parts

Part 4

Two more functions, and their limits

14 minThe exponential and the logarithm, their domains, ranges and graphsThe number both of these functions are built on is never constructed - it is only pinned between 2 and 3, and that statement fixes 0 decimal places. Measured rather than asserted: the search that finds a logarithm contains no test anywhere for a number being negative, and when the identical search is handed four different functions it reports four different domains.14 min · 13 parts

20 minTwo standard limits, the first squeezed out of an inequality and the second reduced to itThe inequality this whole topic rests on is stated and never proved - and the explanation says so rather than pretending otherwise. What it does instead is test it: 40 places inside the interval, 0 failures, against a version of it with one pair of bars missing that fails at 19 of the same 40 and is not even defined at one end.20 min · 17 parts

What you will be able to do

State why an average velocity over an interval is computable and a velocity at an instant is not, in terms of what the quotient needs
State what the left hand limit and the right hand limit of a function at a point are, each in terms of the values the function takes on one side only
State the rule that a limit is read off values near a point and never at it, and justify it from the definition of the one-sided limits
State each of the four parts of Theorem 1 in your own words, including the hypothesis that both limits exist
Derive the limit of xⁿ at a from the product part of Theorem 1 by induction on n
State Theorem 3 and say exactly what it claims about two functions that are ordered on their common domain
State Definition 1 and identify each of its parts: the point, the increment, the quotient and the limiting operation
State each of the four parts of the derivative theorem in §12.5.1 in your own words, including the condition attached to the quotient part
State the power rule for a positive whole exponent
Explain why no rule established so far yields the derivative of sin x, and therefore why the definition must be used
State the domain and the range of the exponential function with base e
State the inequality §12.6 uses, including the interval on which it is asserted and the point excluded from it

What this chapter assumes you already know

Reading a function given by a formula, and evaluating it at decimals
Average speed as distance travelled divided by time taken
That a graph of distance against time has a slope with units of distance per time
The slope of a straight line joining two named points on a curve
Arithmetic with four and five decimal places, done exactly rather than rounded
Reading a piecewise definition: which formula applies on which part of the domain, and which clause owns the boundary point
The modulus function |x| and its graph
Plotting a step or broken graph with filled and hollow endpoint markers

It builds directly on Why writing a pair in order carries information a set cannot, The one-output rule that promotes a relation to a function, A relation is nothing more than a chosen part of the product, Each standard function is pinned down by its picture as much as by its rule, A rule that turns a position number into a term and Shifts by a quarter, a half and a whole turn all fall out of the same two results.

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

"Average velocity over the last stretch just is the velocity at the end."
"The tables prove the velocity is 19.6."
"You could get the exact answer by taking the interval to be zero."
"The left-hand list and the right-hand list are two different phenomena."
"Slope of a chord and average velocity are separate ideas that happen to agree."
"The chapter defines the derivative here."
"If f(a) exists, the limit at a exists."
"If the two clauses are different formulas, the limit must fail."