Suppose f and g are continuous functions such that
g(3) = 2 and the limit as x approaches 3 of [3f(x) + f(x)g(x)] = 15. Find f(3).
3 f(3) + f(3)*2 = 15
5 f(3) = 15
f(3) = 3
Well, let's see... g(3) = 2. I guess you could say it's got a little "2" much going on there. But hey, who am I to judge?
Now, the limit as x approaches 3 of [3f(x) + f(x)g(x)] = 15. Talk about a math problem that loves to party! It's got f(x), g(x), and x all hanging out together. I bet they have some interesting stories to share.
To find f(3), we need to plug in the value of x = 3 into the equation. So let's do some substitution and solve this mystery:
Limit as x approaches 3 of [3f(x) + f(x)g(x)] = 15
[3f(3) + f(3)g(3)] = 15
[3f(3) + f(3)(2)] = 15
Now, we know that g(3) = 2, so let's substitute that in as well:
[3f(3) + f(3)(2)] = 15
3f(3) + 2f(3) = 15
Okay, now we can simplify this equation:
3f(3) + 2f(3) = 15
5f(3) = 15
Divide both sides by 5:
f(3) = 3
So there you have it! f(3) = 3. And just like that, we solved the mystery of the continuous functions f and g. Now they can go back to doing whatever it is continuous functions do in their free time. Maybe they'll even have a little party to celebrate.
To find f(3), we can start by using the given information that the limit as x approaches 3 of [3f(x) + f(x)g(x)] = 15.
Substitute x = 3 into the equation:
[3f(3) + f(3)g(3)] = 15
Since g(3) = 2, we have:
[3f(3) + f(3)*2] = 15
Simplify the equation:
3f(3) + 2f(3) = 15
Combine like terms:
5f(3) = 15
Divide both sides by 5 to solve for f(3):
f(3) = 15/5
f(3) = 3
Therefore, the value of f(3) is 3.
To find f(3), we can use the given information and apply the concept of limits.
We are given that g(3) = 2, so at x = 3, the value of g(x) is 2.
Next, we are given the limit as x approaches 3 of [3f(x) + f(x)g(x)] = 15. We can rewrite this limit as:
lim (x→3) [3f(x) + f(x)g(x)] = 15
Now, let's simplify this expression by factoring out f(x) from the terms inside the brackets:
lim (x→3) f(x) [3 + g(x)]
Since we know that g(3) = 2, we can substitute this value into the expression:
lim (x→3) f(x) [3 + 2]
lim (x→3) f(x) * 5
Now, we have the limit as x approaches 3 of f(x) multiplied by 5 equal to 15. To find f(3), we need to evaluate the limit:
lim (x→3) f(x) * 5 = 15
Dividing both sides of the equation by 5:
lim (x→3) f(x) = 15/5
lim (x→3) f(x) = 3
Therefore, f(3) = 3.