Given that lim x->c f(x)=6 and that lim x->c g(x)= -4, evaluate the following limit. Assume that c is a constant. The x in front of the f is confusing me.
lim [xf(x) + 3 g(x)]^2
x->c
The x in front of the f is any constant you would put in the equation.
Example: x->5 5=c
lim [xf(x) + 3 g(x)]^2
x->5
so, [5*6 + 5*(-4)]^2
You should be able to do the rest.
[c*6 + 3*(-4)]^2 = (6c -12)^2
= 36c^2 - 144c + 144
To evaluate the limit lim [xf(x) + 3g(x)]^2 as x approaches c, we can use limit rules and properties of limits.
Let's break it down step by step:
Step 1: Evaluate the limits of the individual terms.
We know that lim x->c f(x) = 6 and lim x->c g(x) = -4. Now we need to determine the limit of xf(x) and 3g(x) separately.
lim x->c xf(x) = c * (lim x->c f(x))
= c * 6
= 6c
lim x->c 3g(x) = 3 * (lim x->c g(x))
= 3 * (-4)
= -12
So, we have xf(x) approaches 6c and 3g(x) approaches -12 as x approaches c.
Step 2: Substitute the limit expressions back into the original expression.
Now, we substitute the limit expressions we found in the previous step back into the original expression:
lim [xf(x) + 3g(x)]^2
= [lim x->c xf(x) + lim x->c 3g(x)]^2
= [(6c) + (-12)]^2
= (6c - 12)^2
Step 3: Simplify the expression further if possible.
To simplify further, we can expand the square:
(6c - 12)^2 = (6c - 12) * (6c - 12)
= 36c^2 - 72c - 72c + 144
= 36c^2 - 144c + 144
So, the final value of the limit lim [xf(x) + 3g(x)]^2 as x approaches c is 36c^2 - 144c + 144.