Answer
421.8k+ views
Hint: Here first we will calculate the value of the function when the limit of x tends to n and also the value of the given function at \[x = n\] and then apply the condition for continuity of function to prove the given statement.
The condition for continuity of a function f(x) at \[x = a\] is:
\[\mathop {\lim }\limits_{x \to a} f\left( x \right) = f\left( a \right)\]
Complete step-by-step answer:
Let us first consider the given function:-
\[f\left( x \right) = {x^n}\;\;\]
Now we know that the condition for continuity of a function f(x) at \[x = a\] is:
\[\mathop {\lim }\limits_{x \to a} f\left( x \right) = f\left( a \right)\]
And since we have to prove the given function is continuous at \[x = n\]
Hence we have to show:
\[\mathop {\lim }\limits_{x \to n} f\left( x \right) = f\left( n \right)\] by the definition of continuity of a function.
Let us now consider the Left hand side:-
\[LHS = \mathop {\lim }\limits_{x \to n} f\left( x \right)\]
Putting the value of f(x) we get:-
\[LHS = \mathop {\lim }\limits_{x \to n} {x^n}\]
Now evaluating the limit we get:-
\[
LHS = {\left( n \right)^n} \\
\Rightarrow LHS = {n^n} \\
\]
Now let us consider Right hand side:-
\[RHS = f\left( n \right)\]
Putting \[x = n\] in the function we get:-
\[
RHS = {\left( n \right)^n} \\
\Rightarrow RHS = {n^n} \\
\]
Now since \[LHS = RHS\]
Therefore the function is continuous at \[x = n\]
Hence proved.
Note: Students should take a note that a function f(x) is continuous only if \[\mathop {\lim }\limits_{x \to a} f\left( x \right) = f\left( a \right)\]
Also, all functions which are continuous in an interval have continuous graph i.e, the graph does not break at any point in that interval.
A function is said to be discontinuous in an interval if its graph breaks at some points in that interval and also, mathematically,
\[\mathop {\lim }\limits_{x \to a} f\left( x \right) \ne f\left( a \right)\]
The condition for continuity of a function f(x) at \[x = a\] is:
\[\mathop {\lim }\limits_{x \to a} f\left( x \right) = f\left( a \right)\]
Complete step-by-step answer:
Let us first consider the given function:-
\[f\left( x \right) = {x^n}\;\;\]
Now we know that the condition for continuity of a function f(x) at \[x = a\] is:
\[\mathop {\lim }\limits_{x \to a} f\left( x \right) = f\left( a \right)\]
And since we have to prove the given function is continuous at \[x = n\]
Hence we have to show:
\[\mathop {\lim }\limits_{x \to n} f\left( x \right) = f\left( n \right)\] by the definition of continuity of a function.
Let us now consider the Left hand side:-
\[LHS = \mathop {\lim }\limits_{x \to n} f\left( x \right)\]
Putting the value of f(x) we get:-
\[LHS = \mathop {\lim }\limits_{x \to n} {x^n}\]
Now evaluating the limit we get:-
\[
LHS = {\left( n \right)^n} \\
\Rightarrow LHS = {n^n} \\
\]
Now let us consider Right hand side:-
\[RHS = f\left( n \right)\]
Putting \[x = n\] in the function we get:-
\[
RHS = {\left( n \right)^n} \\
\Rightarrow RHS = {n^n} \\
\]
Now since \[LHS = RHS\]
Therefore the function is continuous at \[x = n\]
Hence proved.
Note: Students should take a note that a function f(x) is continuous only if \[\mathop {\lim }\limits_{x \to a} f\left( x \right) = f\left( a \right)\]
Also, all functions which are continuous in an interval have continuous graph i.e, the graph does not break at any point in that interval.
A function is said to be discontinuous in an interval if its graph breaks at some points in that interval and also, mathematically,
\[\mathop {\lim }\limits_{x \to a} f\left( x \right) \ne f\left( a \right)\]
Watch videos on
Prove that the function \[f\left( x \right) = {x^n}\;\;\] is continuous at \[x = n\], where n is a positive integer.
![icon](https://i.ytimg.com/vi_webp/mjxIvla3npI/hqdefault.webp)
Class 12 MATHS | Continuity & Differentiability | NCERT EXERCISE 5 .1 (Question - 4) | Abhishek Sir
Subscribe
likes
10 Views
10 months ago
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)