Prove that the function `f(x)=x^n`is continuous at `x = n`, where n is a positive integer.

Explanation

`f(x) = x^(n)`
For x =n, where n is a positive integer.
`f(n) = x^(n)`
L.H.L. = ` underset (x to n^(-))(lim)f(x) `
Let x = n - h
`rArr n - h to n `
`rArr h to 0`
` underset (h to 0)(lim)f(n -h) `
` underset (h to 0)(lim)(n -h)^(n) = (n -0)^(n) = n^(n)`
R.H.L. = ` underset (h to n^(+))(lim)f(x)`
Let x n + h
`rArr n + h to n `
`rArr h to 0`
` underset (h to 0)(lim)f(0 + h)`
` underset (h to 0)(lim)(n -h)^(n) = (n -0)^(n) = n^(n)`
`:.` L.H.L. = f(n) = R. H. L.
`:.` f(x) is continuous at x = n . Hence proved