Proving Continuity for Two Functions: x+2x^3 and (2x+3)/(x-2)

[EngTechno]

Use the continuity and the properties of limits to show that the function is continuous at the given number.

1. f(x)=( x+2x^3)^4, a=-1

2. f(x)=(2x+3)/(x-2) ,(2,infinity)

 

[HallsofIvy]

Show us what you have done.

In particular what are the "properties of limits" and what is the definition of "continuous function"?

 

[M98Ranger]

1. To prove continuity at a=-1, we need to show that the limit of f(x) as x approaches -1 is equal to f(-1).

First, we can rewrite the function as f(x)=x^4+2x^12. This function is a polynomial, and polynomials are continuous for all real numbers. Therefore, we can say that f(x) is continuous at all real numbers, including -1.

Next, we can use the properties of limits to show that the limit of f(x) as x approaches -1 is equal to f(-1). We know that the limit of a sum is equal to the sum of the limits, so we can split the function into two separate limits:

lim x->-1 x^4 + lim x->-1 2x^12

We can evaluate each limit separately. For the first limit, we can simply plug in -1 for x and get (-1)^4=1. For the second limit, we can factor out a 2x^12 and then plug in -1 for x, giving us 2(-1)^12=2.

Now, we can combine the two limits back together and get 1+2=3. This is the same as f(-1), which we found to be equal to 3 when we plugged in -1 for x in the original function. Therefore, we have shown that the limit of f(x) as x approaches -1 is equal to f(-1), and f(x) is continuous at a=-1.

2. To prove continuity for the function f(x)=(2x+3)/(x-2) at x=2, we can use the same approach as above. We need to show that the limit of f(x) as x approaches 2 is equal to f(2).

First, we can rewrite the function as f(x)=2+3/(x-2). This function is continuous for all real numbers except x=2, as it would result in a division by zero. However, we can use the property of limits that states the limit of a constant is equal to the constant itself. Therefore, we can say that the limit of f(x) as x approaches 2 is equal to 2, which is the same as f(2). This shows that f(x) is continuous at x=2, except for the point x=2 itself.