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Page 1 KEY POINTS ? x c lim ? f(x) = l if and only if ? lim ( ) lim ( ) ? ? ? ? ? ? x c x c f x f x l ? x c lim ? ? ? ? ? ? ? ? where ? is a fixed real number. . ? x c lim ? x n = c n , for all n ? N ? x c lim ? f(x) = f(c), where f(x) is a real polynomial in x. Algebra of limits Let f, g be two functions such that x c lim ? f(x) = l and x c lim ? g(x) = m, then ? x c lim ? [ ? f(x)] = ? x c lim ? f(x) = ? l for all ? ? ? R Page 2 KEY POINTS ? x c lim ? f(x) = l if and only if ? lim ( ) lim ( ) ? ? ? ? ? ? x c x c f x f x l ? x c lim ? ? ? ? ? ? ? ? where ? is a fixed real number. . ? x c lim ? x n = c n , for all n ? N ? x c lim ? f(x) = f(c), where f(x) is a real polynomial in x. Algebra of limits Let f, g be two functions such that x c lim ? f(x) = l and x c lim ? g(x) = m, then ? x c lim ? [ ? f(x)] = ? x c lim ? f(x) = ? l for all ? ? ? R ? x c lim ? [f(x) ± g(x)] = x c lim ? f(x) ± x c lim ? g(x) = l ± m ? x c lim ? [f(x).g(x)] = x c lim ? f(x). x c lim ? g(x) = lm ? x c f(x) lim g(x) ? = x c x c lim f(x) lim g(x) m ? ? ? l , m ? 0 g(x) ? 0 ? x c 1 lim f(x) ? = x c 1 lim f(x) ? = 1 l provided l ? 0 f(x) ? 0 ? x c lim ? [(f(x)] n = n x c lim f(x) ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? = l n , for all n ? N Some important theorems on limits ? x 0 lim ? ? f(x) = x 0 lim ? ? f(–x) ? n n x a x a lim x a ? ? ? = na n – 1 ? x 0 sin x lim x ? =1 where x is measured in radians. ? x 0 x 0 tan x cos x lim 1 Note that lim 1 x x ? ? ? ? ? ? ? ? ? ? ? x 0 1 cos x lim 0 x ? ? ? ? x x 0 e 1 lim 1 x ? ? ? Page 3 KEY POINTS ? x c lim ? f(x) = l if and only if ? lim ( ) lim ( ) ? ? ? ? ? ? x c x c f x f x l ? x c lim ? ? ? ? ? ? ? ? where ? is a fixed real number. . ? x c lim ? x n = c n , for all n ? N ? x c lim ? f(x) = f(c), where f(x) is a real polynomial in x. Algebra of limits Let f, g be two functions such that x c lim ? f(x) = l and x c lim ? g(x) = m, then ? x c lim ? [ ? f(x)] = ? x c lim ? f(x) = ? l for all ? ? ? R ? x c lim ? [f(x) ± g(x)] = x c lim ? f(x) ± x c lim ? g(x) = l ± m ? x c lim ? [f(x).g(x)] = x c lim ? f(x). x c lim ? g(x) = lm ? x c f(x) lim g(x) ? = x c x c lim f(x) lim g(x) m ? ? ? l , m ? 0 g(x) ? 0 ? x c 1 lim f(x) ? = x c 1 lim f(x) ? = 1 l provided l ? 0 f(x) ? 0 ? x c lim ? [(f(x)] n = n x c lim f(x) ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? = l n , for all n ? N Some important theorems on limits ? x 0 lim ? ? f(x) = x 0 lim ? ? f(–x) ? n n x a x a lim x a ? ? ? = na n – 1 ? x 0 sin x lim x ? =1 where x is measured in radians. ? x 0 x 0 tan x cos x lim 1 Note that lim 1 x x ? ? ? ? ? ? ? ? ? ? ? x 0 1 cos x lim 0 x ? ? ? ? x x 0 e 1 lim 1 x ? ? ? ? x e x 0 a 1 lim log a x ? ? ? ? x 0 log(1 x) lim 1 x ? ? ? ? ? ? 1 x x 0 lim 1 x e ? ? ? ? ? s in cos d x x dx ? ? ? cos s in d x x dx ? ? ? ? 2 ta n se c d x x dx ? ? ? 2 cot cos d x ec x dx ? ? ? ? se c se c ta n d x x x dx ? ? ? cos cos .cot d ec x ec x x dx ? ? ? ? 1 . n n d x n x dx ? ? Page 4 KEY POINTS ? x c lim ? f(x) = l if and only if ? lim ( ) lim ( ) ? ? ? ? ? ? x c x c f x f x l ? x c lim ? ? ? ? ? ? ? ? where ? is a fixed real number. . ? x c lim ? x n = c n , for all n ? N ? x c lim ? f(x) = f(c), where f(x) is a real polynomial in x. Algebra of limits Let f, g be two functions such that x c lim ? f(x) = l and x c lim ? g(x) = m, then ? x c lim ? [ ? f(x)] = ? x c lim ? f(x) = ? l for all ? ? ? R ? x c lim ? [f(x) ± g(x)] = x c lim ? f(x) ± x c lim ? g(x) = l ± m ? x c lim ? [f(x).g(x)] = x c lim ? f(x). x c lim ? g(x) = lm ? x c f(x) lim g(x) ? = x c x c lim f(x) lim g(x) m ? ? ? l , m ? 0 g(x) ? 0 ? x c 1 lim f(x) ? = x c 1 lim f(x) ? = 1 l provided l ? 0 f(x) ? 0 ? x c lim ? [(f(x)] n = n x c lim f(x) ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? = l n , for all n ? N Some important theorems on limits ? x 0 lim ? ? f(x) = x 0 lim ? ? f(–x) ? n n x a x a lim x a ? ? ? = na n – 1 ? x 0 sin x lim x ? =1 where x is measured in radians. ? x 0 x 0 tan x cos x lim 1 Note that lim 1 x x ? ? ? ? ? ? ? ? ? ? ? x 0 1 cos x lim 0 x ? ? ? ? x x 0 e 1 lim 1 x ? ? ? ? x e x 0 a 1 lim log a x ? ? ? ? x 0 log(1 x) lim 1 x ? ? ? ? ? ? 1 x x 0 lim 1 x e ? ? ? ? ? s in cos d x x dx ? ? ? cos s in d x x dx ? ? ? ? 2 ta n se c d x x dx ? ? ? 2 cot cos d x ec x dx ? ? ? ? se c se c ta n d x x x dx ? ? ? cos cos .cot d ec x ec x x dx ? ? ? ? 1 . n n d x n x dx ? ? ? ? x x d e e dx ? ? ? .log x x d a a a dx ? ? ? log 1 e d x dx x ? ? ? 0 d constant dx ? Laws of Logarithm ? ? ? log log log e e e A B AB ? ? ? log log log e e e A A B B ? ? ? ? ? ? ? ? ? log log m e e A m A ? ? log 1 0 a ? ? If log then ? ? x B A x B A Let y = f(x) be a function defined in some neighbourhood of the point ‘a’. Let P(a, f(a)) and Q(a + h, f (a + h)) are two points on the graph of f(x) where h is very small and 0 h ? . Page 5 KEY POINTS ? x c lim ? f(x) = l if and only if ? lim ( ) lim ( ) ? ? ? ? ? ? x c x c f x f x l ? x c lim ? ? ? ? ? ? ? ? where ? is a fixed real number. . ? x c lim ? x n = c n , for all n ? N ? x c lim ? f(x) = f(c), where f(x) is a real polynomial in x. Algebra of limits Let f, g be two functions such that x c lim ? f(x) = l and x c lim ? g(x) = m, then ? x c lim ? [ ? f(x)] = ? x c lim ? f(x) = ? l for all ? ? ? R ? x c lim ? [f(x) ± g(x)] = x c lim ? f(x) ± x c lim ? g(x) = l ± m ? x c lim ? [f(x).g(x)] = x c lim ? f(x). x c lim ? g(x) = lm ? x c f(x) lim g(x) ? = x c x c lim f(x) lim g(x) m ? ? ? l , m ? 0 g(x) ? 0 ? x c 1 lim f(x) ? = x c 1 lim f(x) ? = 1 l provided l ? 0 f(x) ? 0 ? x c lim ? [(f(x)] n = n x c lim f(x) ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? = l n , for all n ? N Some important theorems on limits ? x 0 lim ? ? f(x) = x 0 lim ? ? f(–x) ? n n x a x a lim x a ? ? ? = na n – 1 ? x 0 sin x lim x ? =1 where x is measured in radians. ? x 0 x 0 tan x cos x lim 1 Note that lim 1 x x ? ? ? ? ? ? ? ? ? ? ? x 0 1 cos x lim 0 x ? ? ? ? x x 0 e 1 lim 1 x ? ? ? ? x e x 0 a 1 lim log a x ? ? ? ? x 0 log(1 x) lim 1 x ? ? ? ? ? ? 1 x x 0 lim 1 x e ? ? ? ? ? s in cos d x x dx ? ? ? cos s in d x x dx ? ? ? ? 2 ta n se c d x x dx ? ? ? 2 cot cos d x ec x dx ? ? ? ? se c se c ta n d x x x dx ? ? ? cos cos .cot d ec x ec x x dx ? ? ? ? 1 . n n d x n x dx ? ? ? ? x x d e e dx ? ? ? .log x x d a a a dx ? ? ? log 1 e d x dx x ? ? ? 0 d constant dx ? Laws of Logarithm ? ? ? log log log e e e A B AB ? ? ? log log log e e e A A B B ? ? ? ? ? ? ? ? ? log log m e e A m A ? ? log 1 0 a ? ? If log then ? ? x B A x B A Let y = f(x) be a function defined in some neighbourhood of the point ‘a’. Let P(a, f(a)) and Q(a + h, f (a + h)) are two points on the graph of f(x) where h is very small and 0 h ? . Slope of PQ = ? ? ? ? f a h f a h ? ? If 0 h ? , point Q approaches to P and the line PQ becomes a tangent to the curve at point P. ? ? ? ? 0 lim h f a h f a h ? ? ? (if exists) is called derivative of f(x) at the point ‘a’. It is denoted by f’(a). Algebra of derivatives ? ? ? ? ? ? ? ? ? . d d cf x c f x dx dx ? where c is a constant ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? d d d f x g x f x g x dx dx dx ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? . . d d d f x g x f x g x g x f x dx dx dx ? ?Read More
209 videos443 docs143 tests

1. What is the concept of limits in calculus? 
2. How do you find the limit of a function algebraically? 
3. What is the significance of derivatives in calculus? 
4. How do you find the derivative of a function? 
5. What is the relationship between limits and derivatives? 
209 videos443 docs143 tests


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