Suppose f and g are continuous functions such that f(3) = 4 and limx→3[f(x)g(x)+7g(x)] = 12. Find g(3).
Just plug in the limit:
limx→3 [f(x)g(x)+7g(x)] = 12
f(3)g(3) + 7g(3) = 12
4g(3) + 7g(3) = 12
11g(3) = 12
g(3) = 12/11
@oobleck I think you accidentally multiplied the limit by the expression
oops - my bad. time to clean my glasses ...
lol it happens
To find the value of g(3), we can use the given information and apply the concept of limits. Let's break down the problem step by step:
First, let's rewrite the given limit expression using algebraic manipulation:
limx→3[f(x)g(x) + 7g(x)] = 12
Next, we can factor out the common term g(x) from both parts of the expression:
limx→3[g(x)(f(x) + 7)] = 12
Now, we know that the limit of a product is equal to the product of the limits if both limits exist. In this case, we need to find the limit of g(x) as x approaches 3. So, we can rewrite the expression using the limit properties:
limx→3[g(x)] · limx→3[(f(x) + 7)] = 12
Since the limit of (f(x) + 7) is a constant (as x approaches 3), we can say that limx→3[(f(x) + 7)] = f(3) + 7:
limx→3[g(x)] · (f(3) + 7) = 12
Now, we can substitute the given value of f(3) = 4:
limx→3[g(x)] · (4 + 7) = 12
Simplifying the expression gives us:
limx→3[g(x)] · 11 = 12
Now, let's isolate the limit expression. Divide both sides of the equation by 11:
limx→3[g(x)] = 12/11
Therefore, the value of g(3) is 12/11.
To summarize, by using the concept of limits and the given information, we were able to find that g(3) is equal to 12/11.
really? aside from the limit, this is just Algebra I
3(4*g(3) + 7g(3)) = 12
11g(3) = 4
g(3) = 4/11