Limits, Continuity and Differentiability - Evaluations and Examples

Table of Content

What are Limits?

The expression \(\underset{x\to c}{\mathop{\lim }}\,\,f(x)=L\) means that f(x) can be as close to L as desired by making x sufficiently close to ‘C’. In such a case, we say that the limit of f, as x approaches to C, is L.

Neighbourhood of a point

Let ‘a’ be real number and ‘h; is very close to ‘O’ then

Left hand limit will be obtained when x = a – h or x a

Similarly, Right Hand limit will be obtained when x = a + h or x a+

Existence of limit

The limit will exist if the following conditions get fulfilled:

(a) \(\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,\) f(x) =\(\underset{x\to {{a}^{-}}}{\mathop{\lim }}\,\) f(x) i.e. L.H.L = R.H.L

(b) Both L.H.L & R.H.L should be finite.

Examples

  • \(\underset{x\to 1}{\mathop{\lim }}\,\,{{x}^{2}}+1={{1}^{2}}+1=2\)
  • \(\underset{x\to 0}{\mathop{\lim }}\,\,{{x}^{2}}-x={{0}^{2}}-0=0\)
  • \(\underset{x\to 2}{\mathop{\lim }}\,\,\frac{{{x}^{2}}-4}{x+3}=\frac{4-4}{2+3}=0\)
  • In Limits, we have in determinant forms such as \(\frac{0}{0},\frac{\infty }{\infty },0\times \infty ,\infty \times \infty ,{{1}^{\infty }},0{}^\circ ,\infty {}^\circ\)

In these case, we try to simplify the problem into the valid function.

(c) Evaluate \(\underset{x\to \infty }{\mathop{\lim }}\,\frac{x+7\sin x}{-2x+13}\)using sandwich Theorem

Solution: \(-1\le \sin x\le 1\) \(\Rightarrow \,\,-7\le 7\sin x\le 7\) \(\Rightarrow \underset{x\to \infty }{\mathop{\lim }}\,\frac{x-7}{-2x+13}\ge \underset{x\to \infty }{\mathop{\lim }}\,\frac{x+7\sin x}{-2x+13}\ge \underset{x\to \infty }{\mathop{\lim }}\,\frac{x+7}{-2x+13}\)

Watch this Video for More Reference


Use of Expansion in Evaluating limits

Some Important Expansions

  1. \( \log (1+x)=x-\frac{{{x}^{2}}}{2}+\frac{{{x}^{3}}}{3}-\frac{{{x}^{4}}}{4}+…….\)
  2. \({{e}^{x}}=1+x+\frac{{{x}^{2}}}{2!}+\frac{{{x}^{3}}}{3!}-\frac{{{x}^{4}}}{4!}+…….\)
  3. \({{a}^{x}}=1+x\log a+\frac{{{x}^{2}}}{2!}{(log\;a)}^{2}+…….\)
  4. \(\sin x=x-\frac{{{x}^{3}}}{3!}+\frac{{{x}^{5}}}{5!}……\)
  5. \(\cos x=1-\frac{{{x}^{2}}}{2!}+\frac{{{x}^{4}}}{4!}……\)
  6. \(\tan x=x+\frac{{{x}^{3}}}{3}+\frac{2}{15}{{x}^{5}}+……\)

Some Important limits

  1. \(\underset{x\to 0}{\mathop{\lim }}\,\,\,\,\frac{\sin x}{x}=1\)
  2. \(\underset{x\to 0}{\mathop{\lim }}\,\,\,\,\frac{1-\cos x}{{{x}^{2}}}=\frac{1}{2}\)
  3. \(\underset{x\to 0}{\mathop{\lim }}\,\,\,\,\frac{\tan x}{x}=1\)
  4. \(\underset{x\to 0}{\mathop{\lim }}\,\,\,\,\frac{{{e}^{x}}-1}{x}=1\)
  5. \(\underset{x\to 0}{\mathop{\lim }}\,\,\,\,\frac{\log (1+x)}{x}=1\)

Example: \(\underset{x\to 0}{\mathop{\lim }}\,\,\,\,\frac{\sin x-x}{{{x}^{3}}}\)

Solution:

\(\underset{x\to 0}{\mathop{\lim }}\,\,\,\,\,\frac{\sin 2x}{x}\)

Evaluation of Algebric Limits

Direct substitution method

Example: \(\underset{x\to 1}{\mathop{\lim }}\,\,(3{{x}^{2}}+4x+5)\)= 3(1)2 + 4(1) + 5 = 12

Example: \(\underset{x\to 2}{\mathop{\lim }}\,\,\frac{{{x}^{2}}-4}{x+3}=\frac{4-4}{2+3}=\frac{0}{5}=0\)

Factorization method

Example: \(\underset{x\to 2}{\mathop{\lim }}\,\,\frac{{{x}^{2}}-5x+6}{{{x}^{2}}-4}\)

Solution: \(\underset{x\to 2}{\mathop{\lim }}\,\,\frac{(x-2)(x-3)}{(x+2)(x-2)}\Rightarrow \underset{x\to 2}{\mathop{\lim }}\,\frac{x-3}{x+2}\)= \(\frac{-1}{4}\)

Rationalization method

\(\underset{x\to 0}{\mathop{\lim }}\,\,\frac{\sqrt{2+x}-\sqrt{2}}{x}\)

Solution: \(\underset{x\to 0}{\mathop{\lim }}\,\,\frac{\left( \sqrt{2+x}-\sqrt{2} \right)\left( \sqrt{2+x}+\sqrt{2} \right)}{x\left( \sqrt{2+x}+\sqrt{2} \right)}\)

= \(\frac{1}{2\sqrt{2}}\)

 

Theorem: If nd then \(\underset{x\to a}{\mathop{\lim }}\,\frac{{{x}^{n}}-{{a}^{n}}}{x-a}=n{{a}^{n-1}}\)

Example: \(\underset{x\to 2}{\mathop{\lim }}\,\frac{{{x}^{10}}-{{2}^{10}}}{{{x}^{5}}-{{2}^{5}}}\)

 

Solution:

Limit of the form \(0{}^\circ ,\infty {}^\circ\)

Let L = \(\underset{x\to a}{\mathop{lim}}\,{{\left( f(x) \right)}^{g(x)}}\)

Example: \(\underset{x-\infty }{\mathop{\lim }}\,\,\,\,{{x}^{{}^{1}/{}_{4}}}=?\)

Solution: \({{e}^{\log }}\underset{x-\infty }{\mathop{\lim }}\,{{x}^{{}^{1}/{}_{x}}}={{e}^{\underset{x\to 0}{\mathop{\lim }}\,}}\,\,\,\frac{\log x}{x}\)= e° = 1

\(form\,{{1}^{\infty }}\) \(\underset{x\to 0}{\mathop{\lim }}\,{{(1+x)}^{\frac{1}{x}}}=e\,\,\,\,\,\,\,\,\,\,or\,\,\,\,\,\,\,\,\,\,\,\underset{x\to \infty }{\mathop{\lim }}\,{{\left( 1+\frac{1}{x} \right)}^{x}}=e\)

Whenever we have \({{1}^{\infty }}\)case, use \({{e}^{\underset{x\to a}{\mathop{\lim }}\,(f(x)-1)xg(x)}}\)

Example: Evaluate \(\underset{x\to 0}{\mathop{\lim }}\,{{(1+x)}^{cosecx}}\) \(\underset{x\to 0}{\mathop{\lim }}\,{{\left[ {{(1+x)}^{\frac{1}{x}}} \right]}^{cosecx.x}}\)= e1 = e

Continuity

What is Continuity

A continuous function is a function for which small changes in the input results in small changes in the output. Otherwise, a function is said to be discontinuous.

A function f(x) is said to be continuous at x = a if

\(\underset{x\to a}{\mathop{\lim }}\,f(x)=\underset{x\to {{a}^{+}}}{\mathop{\lim }}\,f(x)=f(a)\)

i.e. L.H.L = R.H.L = value of the function at x = a

A function f(x) is said to be discontinuous Function.

Example 1: f(x) = \(\frac{1}{2\sin x-1},\) Discuss the continuity or discontinuity

Solution: Clearly the function will be not defined for \(\sin x=\frac{1}{2}=\sin \frac{\pi }{6}\)

Function is discontinuous for \(x=n\,\pi +{{(-1)}^{n}}\frac{\pi }{6}\)

Example 2: What value must be assigned to K so that the function

\(f(x)=\left\{\begin{matrix} \frac{x^4-256}{x-4} & , & x\neq 4\\ k & , & x=4 \end{matrix}\right.\) is continuous at x=4

Solution: =\(f(4)=\underset{x\to 4}{\mathop{\lim }}\,\,\,\,\,\frac{{{x}^{4}}-256}{x-4}=\underset{x\to 0}{\mathop{\lim }}\,\frac{{{x}^{4}}-{{4}^{4}}}{x-4}={{4.4}^{4-1}}=256\)

Example 3: Discuss the continuity of

(a) Sgn (x3x)

(b) f(x) = \(\left[ \frac{2}{1+{{x}^{2}}} \right]\) , x > 0 [ ]

(a) f(x) = sgn (x3–x)

Here x3 – x = 0 x =0, –1, 1

Hence f(x) is discontinuous at x = 0, 1, –1

(b) \(\frac{2}{1+{{x}^{2}}}\) , x > 0 is a monotonically decimal function

Hence f(x) = \(\left[ \frac{2}{1+{{x}^{2}}} \right],\,\,\,x\ge 0\) is discontinuous,

When \(\frac{2}{1+{{x}^{2}}}\) is on integer

\(\Rightarrow \frac{2}{1+{{x}^{2}}}=1,\,2\,\,\,\,\,\,at\,\,\,\,x=1,0\)

Example 4: Discuss the continuity of f(x) = \(\left\{\begin{matrix} x-2 & x\leq 0\\ 4-x^2 &x>0 \end{matrix}\right.\) at x = 0

Solution: \(\underset{x\to {{0}^{-}}}{\mathop{\lim }}\,f(x)=\underset{x\to {{0}^{-}}}{\mathop{\lim }}\,\,\,x-2=-2\)

Intermediate Value Theorem

If f is continuous on [a, b] and f(a) ≠ f(b), then for any value \(c\in (f(a),f(b))\), there is at least one number in x0 (a, b) for which f(x0) = c

Example: Show that the function f(x) = (xa)2 (xb)2 + x takes the values \(\frac{a+b}{2}\) for some value of \(x\in [a,b]\)

Solution: f(a) = a, f(b) = b. Also f is continuous in [a, b] and \(\frac{a+b}{2}\) \(\in\) \(\in\) [a, b]

Hence using I B I there exists at least one \(c\in [a,b]\) such that f(c) = \(\frac{a+b}{2}\)

Differentiability

Existence of Derivative

Right and left hand derivative

F.H.D: f’(0+) =\(\underset{h\to {{0}^{+}}}{\mathop{\lim }}\,\) \(\frac{f(a+h)-(a)}{h}\)

L.H.D: F’\(({{a}^{-}})\)= \(\underset{h\to {{0}^{-}}}{\mathop{\lim }}\,\) \(\frac{h(a-h)-f(a)}{-h}\)

How can a function fail to be differentiable?

The function f(x) is said to be non-differentiable at x = a if

(a) Both R.H.D & L.H.D exist but not equal

(b) Either or both R.H.D & L.H.D are not finite

(c) Either or both R.H.D & L.H.D do not exist.

Differentiability using first principle of Derivatives

\(f(x)=\left\{\begin{matrix} \frac{sinx^3}{x} & x\neq 0\\ 0 & x=0 \end{matrix}\right.\) at x=0

f(x) = |x|, discuss the differentiability

Solution:

Clearly at x = 0, there is a sharp corner.

Hence the function is non-differentiable

f(x) = |x3|, Discuss the differentiability

f(x) = |x3| = \(\left\{\begin{matrix} \ x^3 & ,x> 0\\ -x^3 & ,x<0 \end{matrix}\right.\)

f’(x) = \(\left\{\begin{matrix} \ 3x^2 & ,x> 0\\ -3x^2 & ,x<0 \end{matrix}\right.\)

f’(0+) = f(0) = 0

Hence f(x) is differentiable at x = 0


Practise This Question

Find the area of triangle whose base is 4 cm long and the height corresponding to the base measures 12 cm.