Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Limits and their Applications

Convergence of a sequence:

A number  l is called the limit of sequence  \left\{ {x_n} \right\} if given  \in > 0 , there exists  m \in N such that  |x_n - l| < \in for  n > m . If  l is the limit of a sequence  \left\{ x_n \right\} , then we write  \lim_{n \to \infty} x_n = l

Definition:

A sequence  \left\{ {x_n} \right\} is said to be convergent if  \lim_{n \to \infty} x_n exists, otherwise it is called divergent.

If  \lim_{n \to \infty} x_n=l, then we say that the sequence  \left\{ x_n \right\} converges to  l .

Note:

A sequence  \left\{ {x_n} \right\} converges to   l iff every sub-sequence of  \left\{ {x_n} \right\} converges to  l .

Theorem:

 \lim_{x \to a} f(x)=l \Longleftrightarrow \lim_{n \to \infty} f(x_n)=l for every real sequence  \left\{ {x_n} \right\} with  x_n \neq a . For any  n \in N and  \lim_{n \to \infty} x_n = a .

Limits of Trigonometry Functions-:

(i)  \lim_{x \to 0} \sin \ x = 0

(ii)  \lim_{x \to 0} \dfrac{\sin  x}{x} = 1 = \lim_{x \to 0} \dfrac{x}{\sin  x}

(iii)  \lim_{x \to 0}~\cos x = 1

(iv)  \lim_{x \to 0} \dfrac{\tan x}{x} = 1 = \lim_{x \to 0} \dfrac{x}{\tan x}

(v)  \lim_{x \to 0} \dfrac{1- \cos x}{x} = 0 = \lim_{x \to 0} \dfrac{x}{1 - \cos x}

(vi)  \lim_{x \to 0} \dfrac{\sin^{-1} x}{x} = 1 = \lim_{x \to 0} \dfrac{x}{\sin^{-1} x}

(vii)  \lim_{x \to 0} \dfrac{\tan^{-1} x}{x} = 1 = \lim_{x \to 0} \dfrac{x}{\tan^{-1} x}

(viii)  \lim_{x \to 0} \dfrac{\sin \ x^o}{x} = \dfrac{\pi}{180}

(ix)  \lim_{x \to a} \dfrac{\sin (x - a)}{x - a} =1

(x)  \lim_{x \to a} \dfrac{\tan (x - a)}{x - a} = 1

(xi)  \lim_{x \to a} \sin^{-1} x = \sin^{-1} a,~ |a| \leq 1

(xii)  \lim_{x \to a} \cos^{-1} x = \cos^{-1} a,~ |a| \leq 1

(xiii)  \lim_{x \to a} \tan^{-1} x = \tan^{-1} a, - \infty < a < \infty

(xiv)  \lim_{x \to \infty} \dfrac{\sin \ x}{x} = \lim_{x \to \infty} \dfrac{\cos x}{x} = 0

(xv) \lim_{x \to \infty} \dfrac{\sin \left( \dfrac{1}{x}\right)}{\dfrac{1}{x}} = 1

(xvi) \lim_{x \to 0} \dfrac{\tan K x}{x} = K

(xvii)  \lim_{x \to 0} \dfrac{1 - \cos x}{x^2} = \dfrac{1}{2}

(xviii)  \lim_{x \to a} \sin \ x = \sin a

\implies \lim_{x \to a} \tan \ x = \tan a if  \cos  a \neq 0

(xix)  \lim_{x \to a} \cos x = \cos a

Limits of Exponential and Logarithmic Functions:

(i)  \lim_{x \to 0} e^x = 1

(ii)  \lim_{x \to 0} \dfrac{e^x - 1}{x} = 1

(iii)  \lim_{x \to 0} \dfrac{a^x - 1}{x} = log_e a

(iv)  \lim_{x \to 0} \dfrac{log (1 + x)}{x} = 1

(v)  \lim_{x \to a} \dfrac{x^n - a^n}{x - a} = na^{n-1}

(vi)  \lim_{x \to 0} (1 + x)^{\frac{1}{x}} =e

(vii)  \lim_{x \to 0} \left( 1 + \dfrac{a}{x} \right)^x = e^a

(viii)  \lim_{x \to \infty} \left( 1 + \frac{a}{x} \right)^x = e^a

(ix)  \lim_{x \to \infty} k^x=\begin{Bmatrix}\infty, & \text{ if }~k > 1 \\0,& \text { if } 0\leq k <1\\1,& \text { if } k=1\end{Bmatrix}

(x) \lim_{x \to \infty} kx=\begin{Bmatrix}\infty, &\text { if } &k > 0 \\-\infty, &\text { if } & k < 0 \\0, & \text{ if } & k=0\end{Bmatrix}

(xi) If  f(x) \leq g(x) \forall x , then  \lim_{x \to a} f(x) \leq \lim_{x \to a} g(x)

(xii) If P=\lim_{x \to a}\left\{ f(x) \right\} ^{g (x)} then,

\begin{aligned} \log_e P&=\lim_{x \to a} g(x) \log f(x)\\\implies & P= _e\lim_{x \to a} g(x) \log \left( f(x) \right) \end{aligned}

Monotonic Sequence:

Definition:

Let  \left\{ x_n \right\} be a real sequence:

(i) The sequence  \left\{ x_n \right\} is said to be monotonic increasing or non-decreasing if  x_n \le x_{n + 1}, \forall n \in N .

(ii) If  x_n < x_{n + 1}, \forall n \in N , then the sequence is strictly increasing.

(iii) If  x_n \ge x_{n + 1}, \forall n \in N , then the sequence is monotonic decreasing.

(iv) If  x_n > x_{n + 1}, \forall n \in N , then the sequence is strictly decreasing.

Bounded Sequence:

(i) A real sequence  \left\{ x_n \right\} is said to be bounded above if there exists a real number  M such that  x_n \leq M, \forall n \in N . The number  M is called an upper bound of the sequence.

(ii)  \left\{ x_n \right\} is said to be bounded below if there exists a real number  m such that  x_n \geq m \forall n \in N , then  m is called lower bound of the sequence.

(iii) If there exists a real number  k > 0 such that   |x_n| \leq K, \forall n \in N , then the sequence  \left\{ x_n \right\} is said to be bounded.

A sequence \left\{ x_n \right\} is said to be Bounded iff it is bounded above and below.

Here m \leq x_n \leq M

Least Upper Bound (LUB) (Supremum)

A real sequence  \left\{ x_n \right\} is said to be bounded above i.e. upper bounds of  \left\{ x_n \right\} exist and denoted as  M , where  x_n \leq M The least upper bound (LUB) or Supremum of a sequence  \left\{ x_n \right\} is said to be the least number exist in the upper bound set of  \left\{ x_n \right\} . i.e.  x_n \le \text{LUB} (M) \le M .

Greatest Lower Bound (GLB) or (Infimum)

A real sequence  \left\{ x_n \right\} is said to be bounded below i.e. lower bounds of  \left\{ x_n \right\} exist and denoted by  m , where  x_n \geq m .

The greatest lower bound (GLB) or infimum of a sequence  \left\{ x_n \right\} is said to be the greatest number exist in the lower bound set of  \left\{ x_n \right\} i.e.  m \geq glb(m) \ge x_n .

Theorem:

(i) A monotonic increasing sequence  \left\{ x_n \right\} bounded above is convergent, i.e.  x_n tends to a limit as  n \to \infty

 \lim_{n \to \infty} x_n = l_1

(ii) A monotonic decreasing sequence  \left\{ x_n \right\} bounded below is convergent, i.e.  x_n tends to a limit as  n \to \infty

 \lim_{n \to \infty} x_n = l_2

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