Suppose f and g are continuous functions such that
g(3) = 4
and
lim
x β 3
[3f(x) + f(x)g(x)] = 35.
Find
f(3).
Well, it seems like f and g are in quite a mathematical relationship here. I guess you could say they're in a "function-ship". π€‘
But let's get serious for a moment. We're given that g(3) = 4, but we want to find f(3). Since f and g are continuous functions, we can use the limit to find f(3).
We're given that:
lim [x β 3] [(3f(x)) + (f(x)g(x))] = 35
Let's break this down a bit. We can rewrite it as:
lim [x β 3] (3f(x)) + lim [x β 3] (f(x)g(x)) = 35
Since g(3) = 4, we can rewrite the second part as:
lim [x β 3] (f(x) * 4)
Now, let's isolate f(x) in the first part:
lim [x β 3] (3f(x)) = 35 - lim [x β 3] (f(x) * 4)
If we simplify the left side, we get:
3 * lim [x β 3] f(x) = 35 - 4 * lim [x β 3] f(x)
Let's call lim [x β 3] f(x) as L. Now our equation becomes:
3L = 35 - 4L
Simplifying further:
7L = 35
Finally, we get:
L = 35/7
L = 5
And there you have it! We found that lim [x β 3] f(x) = 5, which means f(3) = 5. So, f(3) = 5 and we've successfully solved the problem. Hope my "function-ship" didn't confuse you too much! π€‘
To find f(3), we need to make use of the given information and apply the Limit Laws.
From the given limit, we can rewrite it as:
lim(x β 3) [3f(x) + f(x)g(x)] = lim(x β 3) 3f(x) + lim(x β 3) f(x)g(x) = 35.
Since we know g(3) = 4, and g(x) is continuous, we can replace g(x) with its value at x = 3:
lim(x β 3) 3f(x) + lim(x β 3) f(x)g(x) = 3f(3) + f(3)g(3) = 35.
Now, we substitute g(3) = 4 into the equation:
3f(3) + f(3)g(3) = 3f(3) + f(3)(4) = 3f(3) + 4f(3) = 7f(3) = 35.
Simplifying the equation, we have:
7f(3) = 35.
To solve for f(3), we divide both sides of the equation by 7:
f(3) = 35/7 = 5.
Therefore, f(3) = 5.
To find f(3), we can rewrite the limit expression using the given information about g(3) and the limit:
lim
x β 3
[3f(x) + f(x)g(x)] = 35.
Since g(3) = 4, we can substitute 4 for g(x) in the expression:
lim
x β 3
[3f(x) + f(x) * 4] = 35.
Simplifying further:
lim
x β 3
[3f(x) + 4f(x)] = 35.
Combine like terms:
lim
x β 3
[7f(x)] = 35.
Now, we can evaluate the limit:
7f(3) = 35.
Divide both sides of the equation by 7 to solve for f(3):
f(3) = 35/7.
Simplifying:
f(3) = 5.
Therefore, f(3) is equal to 5.
It seems we can assume that
lim(xβ3) f(x) = f(3)
In that case, we have
3f(3) + f(3)*4 = 35
7f(3) = 35
f(3) = 5