Introduction to upper limit definition:
Real function: Let R be the set of all real numbers, and let X and Y be any two nonempty subsets of R.Then,a rule f which associates to each x ε X,a unique real number f(x) ε Y is called a real function from X to Y and we write, f : X ----> Y.f(x) is called the image of x or the value of the function at x.The sets X and Y are called the domain and co domain of f.Also,the set {f(x):x ε X} is called the range of f.
Limit : A function f(x) is said to tend to a limit L as x tends to 'a' if for every ε > 0,however small,there exists a corresponding positive real number δ, such that
l f(x) − L l < ε, Whenever 0 < l x -a l < δ
This is denoted symbolically by `lim_(x->a)` f (x) = L .
It implies that whenever the value of x lies in the open interval (a - δ , a + δ),for δ to be very small, then the value of the function f(x),lies in the open interval (L−ε , L+ε) on the y-axis.
Lower limit and Upper limit : In the limit of a function f(x), as x tends to a, x may approach a from lower side i.e, a < x or from upper side i.e, a > x.The limit of the function f(x),when 'x' tends to 'a' from lower side of a is called lower limit and when 'x' tends to 'a' from upper side of 'a' is called upper limit.
The upper limit and lower limit are represented as shown below.
Upper limit = `lim_(x->a+)` f(x),which implies that x tends to a from upper side of a,i.e, x > a.
lower limit = `lim_(x->a-)` f(x),which implies that x tends to a from lower side of a,i.e, x < a.
Fundamental Theorems of Limit Definition:
1) `lim_(x->a)` { f(x)+g(x)} = {`lim_(x->a)` f(x)} + {`lim_(x->a)` g(x)}.
2) `lim_(x->a)` { f(x)−g(x)} = {`lim_(x->a)` f(x)} − {`lim_(x->a)` g(x)}.
3) `lim_(x->a)` {c . f(x)} = c. {`lim_(x->a)` f(x)}, where c is constant.
4) `lim_(x->a)` {f(x) . g(x)} = {`lim_(x->a)` f(x)} . {`lim_(x->a)` g(x)}.
5) `lim_(x->a)` `(f(x))/(g(x))` = `(lim_(x->a)f(x))/(lim_(x->a)g(x))` , where `lim_(x->a)` g(x) ≠ 0.
6) If f(x)≤ g(x) for all x then `lim_(x->a)` f(x) ≤ `lim_(x->a)` g(x).
Example Problems of Upper Limit
Ex 1: Find the upper limit of `lim_(x->3 )` `(x^2-9)/(x-3) ` ?
Sol : `lim_(x->3+)` `(x^2-9)/(x-3)` = `lim_(x->3+)` `((x-3)(x+3))/(x-3)`
= `lim_(x->3+)` `(x+3)`
= ` 6`
Ex 2: Find the upper limit of `lim_(x->1)` `(x^2-4x+3)/(x-1)` ?
Sol : `lim_(x->1+)` `(x^2-4x+3)/(x-1)` = `lim_(x->1+)` `((x-1)(x-3))/(x-1)`
= `lim_(x->1+)` `(x-3)`
= ` -2`
Is this topic derivative of cotx hard for you? Watch out for my coming posts.
Practice Problems of Upper Limit
Pro 1: Evaluate `lim_(x->1+)` `(6x^2-4x+3)`
Ans: 5
Pro 2: Evaluate `lim_(x->7+)` `(x^2-49)/(x-7)`
Ans: 14
Pro 3: Evaluate `lim_(x->0+)``{(x-2)^2+6}`
Ans: 10
Real function: Let R be the set of all real numbers, and let X and Y be any two nonempty subsets of R.Then,a rule f which associates to each x ε X,a unique real number f(x) ε Y is called a real function from X to Y and we write, f : X ----> Y.f(x) is called the image of x or the value of the function at x.The sets X and Y are called the domain and co domain of f.Also,the set {f(x):x ε X} is called the range of f.
Limit : A function f(x) is said to tend to a limit L as x tends to 'a' if for every ε > 0,however small,there exists a corresponding positive real number δ, such that
l f(x) − L l < ε, Whenever 0 < l x -a l < δ
This is denoted symbolically by `lim_(x->a)` f (x) = L .
It implies that whenever the value of x lies in the open interval (a - δ , a + δ),for δ to be very small, then the value of the function f(x),lies in the open interval (L−ε , L+ε) on the y-axis.
Lower limit and Upper limit : In the limit of a function f(x), as x tends to a, x may approach a from lower side i.e, a < x or from upper side i.e, a > x.The limit of the function f(x),when 'x' tends to 'a' from lower side of a is called lower limit and when 'x' tends to 'a' from upper side of 'a' is called upper limit.
The upper limit and lower limit are represented as shown below.
Upper limit = `lim_(x->a+)` f(x),which implies that x tends to a from upper side of a,i.e, x > a.
lower limit = `lim_(x->a-)` f(x),which implies that x tends to a from lower side of a,i.e, x < a.
Fundamental Theorems of Limit Definition:
1) `lim_(x->a)` { f(x)+g(x)} = {`lim_(x->a)` f(x)} + {`lim_(x->a)` g(x)}.
2) `lim_(x->a)` { f(x)−g(x)} = {`lim_(x->a)` f(x)} − {`lim_(x->a)` g(x)}.
3) `lim_(x->a)` {c . f(x)} = c. {`lim_(x->a)` f(x)}, where c is constant.
4) `lim_(x->a)` {f(x) . g(x)} = {`lim_(x->a)` f(x)} . {`lim_(x->a)` g(x)}.
5) `lim_(x->a)` `(f(x))/(g(x))` = `(lim_(x->a)f(x))/(lim_(x->a)g(x))` , where `lim_(x->a)` g(x) ≠ 0.
6) If f(x)≤ g(x) for all x then `lim_(x->a)` f(x) ≤ `lim_(x->a)` g(x).
Example Problems of Upper Limit
Ex 1: Find the upper limit of `lim_(x->3 )` `(x^2-9)/(x-3) ` ?
Sol : `lim_(x->3+)` `(x^2-9)/(x-3)` = `lim_(x->3+)` `((x-3)(x+3))/(x-3)`
= `lim_(x->3+)` `(x+3)`
= ` 6`
Ex 2: Find the upper limit of `lim_(x->1)` `(x^2-4x+3)/(x-1)` ?
Sol : `lim_(x->1+)` `(x^2-4x+3)/(x-1)` = `lim_(x->1+)` `((x-1)(x-3))/(x-1)`
= `lim_(x->1+)` `(x-3)`
= ` -2`
Is this topic derivative of cotx hard for you? Watch out for my coming posts.
Practice Problems of Upper Limit
Pro 1: Evaluate `lim_(x->1+)` `(6x^2-4x+3)`
Ans: 5
Pro 2: Evaluate `lim_(x->7+)` `(x^2-49)/(x-7)`
Ans: 14
Pro 3: Evaluate `lim_(x->0+)``{(x-2)^2+6}`
Ans: 10
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