Question

Use the definition of continuity and the properties of limits to show that the function f(x) = x^2 + 5(x - 2)^7 is continuous at x = 3

Answer 1

Answer:

See below.

Step-by-step explanation:

First check that f(x) has a real value at x = 3:

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

=  9 + 5 * 1^7

=  14,

So the first condition is met.

Now we check if limit as x approaches 3 exists.

As x approaches 3 from below f(x) approaches 14 and at x = 3 = 14.

As x approaches 3 from above f(x) approaches 14 and at x = 3 = 14.

These 3 conditions  shows that f(x) is continuous at x = 3.