CHAPTER 1 : Introduction Limits and Continuity

What is Calculus? Mathematics of motion and change Whenever there is motion or growth, whenever there is variable forces at work Learning calculus is not the same as learning arithmetic, algebra, and geometry. In those subjects, you learn primarily how to calculate with numbers, how to simplify algebraic expressions and calculate with variables, and how to reason with points, lines, and figures in the plane. Calculus involves those techniques and skills but develops others as well, with greater precision and at deeper level.

Learning Calculus Read the text Do the homework Use calculators and computers Try writing your own notes and short descriptions Rewrite definitions etc to increases persistency in understanding

Limits

Overview One of the chapters that connects Calculus and Algebra and Trigo. Straightforward calculations and can be solved by simple substitution. Limits – describe the way a function changes Some functions vary continuously, small changes in x produce only small changes in f(x). Other functions can have values that jump or vary erratically.

Speed A rate of change A moving body’s average speed over any particular time interval is the amount of distance covered during the interval divided by the length of the interval. Example: A rock falls of a tall cliff. What is its average speed during :- (i) First 2 seconds (t = 2 sec) of fall (ii) At time t = 2 seconds

Gradient of secants as approaching limit

f(x 2 ) x2x2 f(x 1 ) x1x1 f(x) Average Rate of Change ∆ y ∆ x

Limits f(x) approaches arbitrarily closer to a h 1 when x approaches h 2

Informal Limits f(x), x=x 0, x 0 ≠x 0 If f(x) close to L for all x values close to x 0 f(x) approaches limit L as x approaches x 0 “informal” because phrases like arbitrarily close or sufficiently close are imprecise

Rules for Finding Limits Limits can be assessed algebraically, using arithmetic and rules Theorem 1: Limit Rules The following rules hold if and where L, M, c and k are real numbers

Theorem 1 : Limit Rules

Theorem 2

Theorem 3

Solving Limit Questions THREE step/method rule STEP 1: Substitute/Replace x value If unsuccessful STEP 2: Factorise to cancel nominator If unsucceful STEP 3: Conjugation REMEMBER  There is no such thing as an undefined limit! 0 or any other number is still a limit