Define f, g : R → R by f (x) = (x − 2)2 andg(x) =1 if x > 0,0 if x = 0,−1 if x < 0.Calculatelimx→2 g(f (x)) and g limx→2 f (x)

First, let's calculate the limit of f(x) as x approaches 2.

lim(x→2) f(x) = lim(x→2) (x - 2)^2

As x approaches 2, (x - 2)^2 approaches 0. So, lim(x→2) f(x) = 0.

Now, let's calculate g(f(x)) as x approaches 2.

g(f(x)) = g((x - 2)^2)

As we found above, as x approaches 2, (x - 2)^2 (which is f(x)) approaches 0. So, we need to find the value of g(x) when x = 0.

From the definition of g(x), g(0) = 0. So, lim(x→2) g(f(x)) = 0.

Finally, let's calculate g(lim(x→2) f(x)).

We found above that lim(x→2) f(x) = 0. So, we need to find the value of g(x) when x = 0.

From the definition of g(x), g(0) = 0. So, g(lim(x→2) f(x)) = 0.

So, both lim(x→2) g(f(x)) and g(lim(x→2) f(x)) are equal to 0.