Limit MATHEMATICS LIMIT CRASH COURSE Limit of a

Limit MATHEMATICS LIMIT CRASH COURSE

Limit of a function f(x) is said to exist as, x ® a when, f(a - h) (Left hand limit) = f (a + h) = a. f. q (a finite quantity say M). (Right hand limit) = afq Note: # We are not interested in knowing about what happens at x = a. # Also note that if L. H. L. & R. H. L. are both tending towards ' ¥ ' or ‘– ¥’ then it is said to be infinite limit. # Remember, Þ x ¹ a

Question (i) Find (iii) Find

Limit Indeterminant Forms: , , 0 × ¥, ¥ - ¥, ¥ 0, 00, and 1¥. Note : (i) ‘ 0‘ doesn't means exact zero but represent a value approaching towards zero similarly to ‘ 1' and infinity. (ii) ¥+¥=¥ (iii) ¥x¥=¥ (iv) (a/ ¥) = 0 (if a is finite)

Limit Method of Removing Indeterminancy : To evaluate a limit, we must always put the value where ‘x' is approaching to in the function. If we get a determinate form, then that value becomes the limit otherwise if an indeterminant form comes then apply one of the following methods: (i) Factorisation (ii) Rationalisation or double rationalisation (iii) Substitution (iv) Using standard limits (v) Expansions of functions.

Limit (i) Factorization Method Indeterminant Forms: We cancel out the factors which are leading to indeterminacy And find the limit of the remaining expression.

Question Evaluate (A)10 (B) 11 (C) 12 (D) 13 :

Limit (ii) Rationalization / Double Rationalization : We can rationalize the irrational expression by multiplying with their conjugates to remove the indeterminancy.

Question Evaluate (A) (B) (C) – 1 (D) 1 :

Question Evaluate (A) 1 (B) 2 (C) 3 (D) 4 :

Question Evaluate (A) (B) (C) (D) :

Limit (iii) Fundamental Theorems on Limits : Let f (x) = & g (x) = m. If & m exists, then: (i) {f(x) ± g (x)} = l ± m (ii) {f(x). g(x) } = l. m (iii) = (iv) k f(x) = k (v) f[g(x)] = g(x) = m. , provided m ¹ 0 f(x) ; where k is a constant. = f (m); provided f is continuous at

Limit (iv) Standard Limits : (a) =1= = = [Where x is measured in radians ] (b) (c) (d) (e) (1 + x)1/x = e ; = 1; =1 = nan – 1. =e = logea, a > 0

Question

Question Evaluate (A) 1 (B) 2 (C) 3 (D) 4 :

Limit When x ® ¥ Since x ® ¥ Þ ® 0 hence in this type of problem we express most of the part of expression in terms of and apply ® 0.

Question Evaluate (A) 0 (B) 1 (C) 2 (D) 3 :

Question Evaluate (A) (B) (C) (D) :

Limits Using Expansion : (i) (iii) (iv) (v) ln (1+x) =

Limit (vi) (viii) ln (1+x) = (ix) (x) for |x| < 1, n Î R (1 + x)n = 1 + nx + + +. . . ¥

Question Evaluate (A) (B) (C) (D) :

Question Evaluate (A) (B) (C) (D) :

Question Evaluate (A) (B) (C) (D) :

Question Evaluate (A) (B) (C) (D) :

Limits of form 1¥, 00, ¥ 0 All these forms can be convered into form in the following ways

Question Evaluate (A) e– 8 (B) e– 1 (C) e 2 (D) e-2/p :

Question Evaluate (A) e– 8 (B) e– 1 (C) e 2 (D) e-2/p :

Question Evaluate (A) e– 8 (B) e– 1 (C) e 2 (D) e-2/p :

Question Evaluate (A) 1 (B) 2 (C) 3 (D) 4 :

Limit Sandwich Theorem or Squeeze Play Theorem: If f(x) £ g(x) £ h(x) " x & =l= then lim g(x) = l.

Question Evaluate Where [ ] denotes the greatest integer function. (A) (B) (C) (D)

Question

Limit Some Important Notes (i) : (ii) As x® ¥, ln x increases much slower than any (+ve) power of x where e x increases much faster than (+ve) power of x (iii) (iv) If (1 - h) = 0 & (1 + h)n ® ¥ where h > 0. n f(x) = A > 0 & f (x) = B (a finite quantity) then; [f(x)]f(x) = ez where z = f (x). ln[f(x)] = e. Bin. A = AB

Question Built in Limit Concept:

Question

Question Evaluate (A) 0 (B) 2 (C) 3 (D) 4 :

Question Let f be a differentiable function such that f'(x) = 7 – f(1) ¹ 4. Then (A) exists and equals 0 (B) exists and equals (C) does not exist. (D) exists and equals 4. and

Question For each x Î R, let [x] be the greatest integer less than or equal to x. Then (A)1 (B) sin 1 (C) – sin 1 (D) 0 is equal to:

Question If then is __

Question Let f(x)= 5 – |x – 2| and g(x) = |x + 1|, x Î R. If f(x) attains maximum value at a and g(x) attains minimum value at b, then is equal to (A) 3/2 (B) – 3/2 (C) – 1/2 (D) 1/2

Question If (A)-7 (B) -4 (C) 1 (D) 5 then a + b is equal to :

Question If (A) (B) (C) (D) , then k is :

Question Let [x] denote the greatest integer less than or equal to x. then : (A)does not exist (B) equals 0 (C) equals p + 1 (D) equals p

Question If integer function then where [. ] denotes greatest =. . .

Question Let e denote the base of the natural logarithm. The value of the real number for which the right hand limit non-zero real number, is____ is equal to a

Question Let f : R ® R be a differentiable function satisfying f'(3) + f'(2) = 0. Then (A)e 2 (B) 1 (C) e-1 (D) e is equal to:

Question equals: (A) 4 (B) (C) (D)