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Limits, Continuity

Lecture on Limit
Content covered: 

Introduction to Limit of a function

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Lecture on Formula and Results
Content covered: 

Important Formula and results.

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Lecture on Indeterminate Form
Content covered: 

Indeterminate Forms

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\(\lim_{x \to \infty} \frac{x - \sin{x}}{x - \cos{x}}\) is equal to

(a)    1

(b)    -1

(c)    \(\infty\)

(d)    -\(\infty\)

GATE 2007,  ME

Find     \(\lim_{x \rightarrow 0} \frac{e^x - (1+x+ \frac{x^2}{2})}{x^3}\)

Lecture on Continuity
Content covered: 

Continuity of a function

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The function \(f(x) = \frac{x^3 - 8}{x^2-4}\)  is undefined at  \(x= 2\) . Redefine the function to make it continuous at \(x=2\).

Example on Limits

\(\lim_{x\to4} {sin(x - 4) \over x - 4} = ?\) 

Let ,

x - 4 = h

∴ when, \(x \to 4 \) then, \(h \to 0\)

Now, it becomes 

\(\lim_{x\to4} {sin(x - 4) \over x - 4} = \lim_{h\to0} {sin(h) \over h} = 1\) 

Hence, Ans = 1

 

 

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Example on Continuity

Which one of the following functions is continuous at x = 3 ?

 

(A)\(f(x) = \left\{ \begin{array}{ll} 2 & x = 3 \\ x -1 & x > 3\\ \frac{x+3}{3} & x < 3 \\ \end{array} \right. \)

 

(B) \(f(x) = \left\{ \begin{array}{ll} 4 & x = 3 \\ 8-x & x \neq 3\\ \end{array} \right. \)

 

(C) \(f(x) = \left\{ \begin{array}{ll} x+3 & x \leq 3 \\ x -4 & x > 3\\ \end{array} \right. \)

 

(D) \(f(x) = \left\{ \begin{array}{ll} \frac{1}{x^3 - 27} & x \neq 3 \\ \end{array} \right. \)

Things you need to know

For a function to be continuous at 3, the Left Hand Limit (LHL) = Right Hand Limit = \(f(3)\)

 

(a) 

\(f(3) = 2\)

LHL = \(\lim_{h \to 0} f(3 - h)\)  = \(\lim_{h \to 0} {\frac{3-h+3}{3}}\) = 2

RHL = \(\lim_{h \to 0} f(3 + h)\) = \(\lim_{h \to 0} {3 + h -1}\) = 2

∵ LHL = RHL = \(f(3)\)

∴ continuous at x=3.

(b)

\(f(3) = 4\)

LHL = \(\lim_{h \to 0} f(3 - h)\) = \(\lim_{h \to 0} {8 - (3 - h)}\)= 5

RHL = \(\lim_{h \to 0} f(3 + h)\) = \(\lim_{h \to 0} {8 - (3 + h)}\)  = 4

∵ LHL = RHL \(\neq f(3)\)

∴ Not continuous at x = 3

 

(c)

\(f(3) = 3 + 3 = 6\)

LHL = \(\lim_{h \to 0} f(3 - h)\) = \(\lim_{h \to 0} {3+h+3} = 6\)

RHL = \(\lim_{h \to 0} f(3 + h)\) = \(\lim_{h \to 0}{3 + h - 4} = -1\)

∵ \(f(3) = LHL \neq RHL\)

∴ Not continuous at x = 3.

 

(d) 

\(f(3)\) is not defined , therefore not continuous at x = 3

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