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

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The discussion focuses on proving the continuity of two functions: f(x) = (x + 2x^3)^4 at a = -1 and f(x) = (2x + 3)/(x - 2) at x = 2. For the first function, it is established that since it is a polynomial, it is continuous for all real numbers, including at -1, where the limit equals the function value. In the second case, the function is shown to be continuous everywhere except at x = 2 due to a division by zero, but the limit as x approaches 2 is evaluated to be 2, which matches the function's value at that point. The properties of limits and the definition of continuity are key to these proofs. Overall, the continuity of both functions is effectively demonstrated through limit evaluation.
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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)
 
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Show us what you have done.

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


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.
 
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