Find all values of a, B so that the function f(x)=1 if x>3, f(x)=ax-10 if x<3, and f(x)=B otherwise is continuous everywhere. I think a is continuous everywhere since ax-10 is a polynomial, and b is continuous at x=3.
You have the right idea. But a must be chosen so that
lim (ax-10) = 1 as x->3+
f(x) is defined every where except at x=3, so define it to be 1 there, since the limit from both sides is 1. That is,
f(3) = B
a = 11/3
B = 1
To determine the values of a and B for which the function f(x) is continuous everywhere, we need to ensure that the function is continuous at x = 3.
Recall that a function is continuous at a specific point if it satisfies these three conditions:
1. The function is defined at that point.
2. The limit of the function as x approaches that point exists.
3. The value of the function at that point matches the limit.
Let's analyze the function f(x) at x = 3 by considering the left-hand limit, the right-hand limit, and the value of the function at x = 3.
1. Left-hand limit:
lim(x->3-) (ax - 10)
As x approaches 3 from the left side, the function f(x) becomes ax - 10. Since this is a polynomial, it is continuous everywhere. Therefore, the left-hand limit exists.
2. Right-hand limit:
lim(x->3+) 1
As x approaches 3 from the right side, the function f(x) becomes 1. This is a constant function, and it is continuous everywhere. Therefore, the right-hand limit exists.
Now, let's find the value of f(x) at x = 3:
f(3) = B
To ensure that f(x) is continuous at x = 3, we need the following conditions to hold:
1. The left-hand limit is equal to the value of the function: lim(x->3-) (ax - 10) = B
2. The right-hand limit is equal to the value of the function: lim(x->3+) 1 = B
From the left-hand limit, we have:
lim(x->3-) (ax - 10) = B
Using the limit definition, we substitute x = 3 into the expression:
a(3) - 10 = B
3a - 10 = B
From the right-hand limit, we have:
lim(x->3+) 1 = B
Since the right-hand limit is a constant function, the value is already B.
Now, to summarize:
- The value of a does not affect the continuity of f(x) at x = 3. Since ax - 10 is a polynomial, it is continuous everywhere. Therefore, any value of a will satisfy the continuity condition.
- The value of B needs to be the same for both the left-hand limit and the right-hand limit. Thus, B should satisfy the equation 3a - 10 = B.
In conclusion:
- The value of a can be any real number because ax - 10 is a continuous function for any value of a.
- The value of B can be found by solving the equation 3a - 10 = B. Any value of B that satisfies this equation will ensure the continuity of f(x) at x = 3.