Question

Use the definition of continuity and the properties of limits to show that the function f(x)=x sqrtx/(x-6)^2 is continuous at x = 36.

Answer 1

The function is continuous at x = 36

Explanation:

From the question we are told that

The function is
`f(x) = x * \sqrt{ (x)/( (x-6) ^2 ) }`

The point at which continuity is tested is x = 1

Now from the definition of continuity ,

At function is continuous at k if only

`\lim_(x \to k)f(x) = f(k)`

So

`\lim_(x \to 36)f(x) = \lim_(n \to 36)[x * \sqrt{ (x)/( (x-6) ^2 ) }]`

`= 36 * \sqrt{ (36)/( (36-6) ^2 ) }`

`= 7.2`

Now

`f(36) = 36 * \sqrt{ (36)/( (36-6) ^2 ) }`

`f(36) = 7.2`

So the given function is continuous at x = 36

because

`\lim_(x \to 36)f(x) = f(36)`