Question

Use the definition of continuity and the properties of limits to show that the function

f(x) = x2 + 5(x - 2)7 is continuous at x = 3.

Answer 1

Applied the definition and the limit.

They had the same result, so the function is continuous.

Explanation:

At function f(x) is continuous at x = a if:

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

In this question:

`f(x) = x^(2) + 5(x-2)^(7)`

At x = 3.

`\lim_(x \to 3) x^(2) + 5(x-2)^(7) = 3^(2) + 5(3-2)^(7) = 14`

`f(3) = 3^(2) + 5(3-2)^(7) = 14`

Since
`\lim_(x \to 3) f(x) = f(3)`, f(x) is continuous at x = 3.