discuss the continuity of the following funtion at the indicated point

f(x) ={1-xn/1-x ,x is not = 1

{n-1, x=1

n is the element of N

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discuss the continuity of the following funtion at the indicated
point
f(x) ={1-xn/1-x ,x is not = 1
{n-1, x=1
n is the element of N

Gursheen kaur. · Accepted Answer

Now,

LHL (x=1)

$\lim_{x \to 1^{-}} \frac{1 - x^{n}}{1 - x} = \lim_{x \to 1^{-}} \frac{(1 - x) [1 + x + x^{2} + ......... (x^{n} + 1)]}{1 - x} = \lim_{x \to 1^{-}} 1 + x + x^{2} + ......... (x^{n} + 1)$

RHL (x=1)

$\lim_{x \to 1^{+}} \frac{1 - x^{n}}{1 - x} = \lim_{x \to 1^{+}} \frac{(1 - x) [1 + x + x^{2} + ......... (x^{n} + 1)]}{1 - x} = \lim_{x \to 1^{+}} 1 + x + x^{2} + ......... (x^{n} + 1)$

⇒LHL(x=1) = RHL (x=1)

Therefore,

$\lim_{x \to 1} f (x) = \lim_{x \to 1} \frac{1 - x^{n}}{1 - x} = \lim_{x \to 1} 1 + x + x^{2} + ... + (x^{n} + 1) a n d f (1) = n - 1 \Rightarrow \lim_{x \to 1} f (x) \neq f (1)$

⇒The function is discontinuous at x = 1.

discuss the continuity of the following funtion at the indicated point f(x) ={1-xn/1-x ,x is not = 1 {n-1, x=1 n is the element of N

discuss the continuity of the following funtion at the indicated point

f(x) ={1-xn/1-x ,x is not = 1

{n-1, x=1

n is the element of N