Tiêu đề LIMITS & DERIVATIVES (Q).pdf ✅

\\ Chapter – 1 The concept of limit is used to discuss the behaviour of a function close to a certain point. e.g., 1 1 ( ) 2    x x f x Clearly the function is not defined at x = 1, but for values close to x = 1 the function can be written as f(x) = x + 1 As x approaches 1 (written as x  1), f(x) approaches the value 2 (i.e., f(x)  2) we write this as lim ( ) 2 1   f x x It must be noted that it is not necessary for the function to be undefined at the point where limit is calculated. In the above example lim ( ) 2 f x x is the same as the value of function at x = 2 i.e., 3. Informally, we define limit as: Let f(x) be defined on an open interval about x0, except possibly at x0 itself. If f(x) gets arbitrarily close to L for all x sufficiently close to x0, we say that f approaches the limit L as x approaches x0, and we write f x L x x   lim ( ) 0 Sometimes, functions approach different values as x-approaches x0 from left and right. By left we mean x < x0 and right means x > x0. This is written as x   x0 and x   x0 respectively. e.g., f (x) = [x] (greatest integer function) For any integer n, lim ( )  1   f x n x n ...(i) and f x n x n    lim ( ) ...(ii) In such cases we say that lim f(x) xn does not exist. The limit in (i) is said to be the left hand limit (L.H.L.) at x = n and that in (ii) is called the right hand limit (R.H.L.) at x = n. 1.1 TANGENT LINE AND SLOPE PREDICTOR In elementary geometry the line tangent to a circle at a point P is defined as the straight line through P that is perpendicular to the radius OP to the point P. O P A general graph y  fx has no radius for as to use, but the line tangent to the graph at the point P should be the straight line through P that has in some sense the same direction at P as the curve itself. Because a line’s “direction” is determined by its slope. Let us take an example: LIMIT OF A FUNCTION 1 THEORY CONTENT OF LIMITS AND DERIVATIVES
(a) Determine the slope of the line L tangent to the parabola 2 y  x at the point   2 a, a . K Q P L (a, a2 ) (b, b 2 ) We can’t immediately calculate the slope L, because we know the co-ordinate of only one point   2 a, a of L. Hence we begin with another line whose slope we can compute. In figure secant line K that passes through the point P and a very close point   2 Q b, b of the parabola 2 y  x . Let us write h  x  b  a for the difference of x-co-ordinate of P and Q. (the notation x is an increment, or change in the value of x). The co-ordinates of Q is given by b  a  h and   2 2 b  a  h Hence the difference in y-co-ordinates of P and Q is   2 2 2 2 y  b  a  a  h  a Because P and Q are two different points, we can use the definition of slope to calculate the slope mPQ of the secant line K through P and Q. If you change the value of h = x, then also change the line K and thereby change its slope. Therefore mPQ depends on h. x y mPQ    =                   h ah h a h a a h a 2 2 2 2 =   h h 2a  h because h is non-zero we can cancel it out and it becomes = 2a  h Now, if Q moves along the curve closer and closer to the point P. The line K still passes through P and Q. It is very close to P then h approaches zero, the secant line K comes closer to coinciding with the tangent line L. Out idea is to define the tangent line L as the limiting position of the secant line K. h: approaches zero Q: approaches P, and so K: approaches L, mean while the slope of K approaches the slope of L. As the number h approaches zero, what values does the slope mPQ  2a  h approaches we can state this question of the “limiting value” of 2a + h by writing  a h h   lim 2 0 Here “lim” is an abbreviation for the word “limit” and h  0 is an abbreviation for the phrase “h approaches zero”, then we can give the answer that what is the limit of 2a + h as h approaches zero. For any specific value of a we can investigate this question numerically by calculating values of 2a  h with values of h that become closer and closer to zero, such as the values h = –0.1, h = –0.01, h = –0.001, h = –0.0001, ......, or the values h = 0.5, h = 0.1, h = 0.05, h = 0.01, ...... For instance, the tables of values (in figure) indicate that with a = 2 and a = –4, we should conclude that lim 2  2 0    h h and lim  4  4 0      h h More generally, it seems clear from the table in figure that m  a h a h PQ h lim lim 2 2 0 0      ...(i) h 2+h 0.1 2.1 0.01 2.01 0.001 2.001 h –4+h 0.5 –3.5 0.1 –3.9 0.05 –3.95 0.01 –3.99
As h  0 (first column), 2 + h approaches 2 (second column) As h  0 (first column), –4 + h approaches –4 (second column) As h  0 (first column), 2a + h approaches 2a (second column) This, finally, answers our original question: The slope m = mPQ of the line tangent to the parabola 2 y  x at the point   2 a, a is given by m  2a ...(ii) The formula in equation (ii) is a “slope predictor” for (lines tangent to) the parabola 2 y  x . Once we know the slope of the line tangent to the curve at a given point of the curve, we can then use the point-slope formula to write an equation of this tangent line. We define the slope m of the line tangent to the graph y  fx at the point Pa, fa to be     h f a h f a m h    0 lim If a  h is x h  x  a . We set that x approaches a as h approaches 0. So,               x a f x f a m x a lim 2.1 CONSTANT LAW If fx  C , where C is a constant [so fx is a constant function], then fx C C x a x a     lim lim ...(i) 2.2. SUM LAW If both of the limits fx L x a   lim and gx M x a   lim exist, then fx gx  fx  gx L M x a x a x a         lim[ ] lim lim ...(ii) (The limit of a sum is the sum of the limits, the limit of a difference is the difference of the limits). 2.3 PRODUCT LAW If both of the limits fx L x a   lim and gx xa lim = M exist, then fxgx  fx gx LM x a x a x a      lim[ ] lim lim ...(iii) (The limit of a product is the product of the limits). h 2a+h 0.01 2a + 0.01 0.001 2a + 0.001     0 2a THE LIMIT LAWS 2
2.4 QUOTIENT LAW If both of the limits fx L x a   lim and gx M x a   lim exist and if M  0 , then         M L g x f x g x f x x a x a x a      lim lim lim ...(iv) (The limit of a quotient is the quotient of the limits, provided that the limit of the denominator is not zero.) 2.5 ROOT LAW If n is a positive integer and if a  0 for even values of n, then n n x a x  a  lim ...(v) The case n = 1 of the root law is obvious x a x a   lim Examples (ii) and (iii) show how the limit laws can be used to evaluate limits of polynomials and rational functions. Example 1: lim  2 4 lim  lim 2  lim 4 3 3 2 3 2 3         x x x x x x x x = lim  2lim  lim 4 3 2.3 4 19 2 3 3 2 3       x x x x x Example 2:   lim 2 4 lim 2 5 2 4 2 5 lim 2 3 3 2 3           x x x x x x x x x = 19 11 3 2.3 4 2.3 5 2     Note: In examples 1 and 2, we systematically applied the limit laws until we could simply substitute 3 for 3 lim x x at the final step. To determine the limit of a quotient of polynomials, we must verify before this final step that the limit of the denominator is not zero. If the denominator limit is zero, then the limit may fail to exist. 2.6 SOME IMPORTANT FORMULAE (i) 1 sin lim 0   x x x (ii) 1 tan lim 0   x x x (iii) 1 sin lim 1 0    x x x (iv) 1 tan lim 1 0    x x x (v) 1 1 lim 0    x e x x (vi) a x a x x ln 1 lim 0    (vii) 1 lim      n n n x a na x a x a (viii)  x e x x    1/ 0 lim 1 (ix) e x x x          1 lim 1 (x)   1 ln 1 lim 0    x x x Illustration 1