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)
Question
Define f, g : R → R by
f (x) = (x − 2)²
and
g(x) =
\begin{cases}
1 & \text{if } x > 0,
0 & \text{if } x = 0,
-1 & \text{if } x < 0.
\end{cases}
Calculate
and
Solution
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)
We know that as x approaches 2, (x - 2)^2 (which is f(x)) approaches 0. So, we need to find the value of g at 0.
From the definition of g, we know that g(0) = 0. So, g(f(x)) = 0 as x approaches 2.
Therefore, lim(x→2) g(f(x)) = 0.
Finally, let's calculate g(lim(x→2) f(x)).
We already know that lim(x→2) f(x) = 0. So, we need to find the value of g at 0.
From the definition of g, we know that g(0) = 0.
Therefore, 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.
Similar Questions
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)
Use the graphs of f and g in the accompanying figure tofind the limits that exist. If the limit does not exist, explainwhy.(a) limx → 2 [f(x) + g(x)]
Fie functiile f, g : R −→ R, definite prin f (x) = x − 2, g(x) = 2 − x, ∀x ∈ R(a) Sa se calculeze (f ◦ g)(1)
Use the graph to find the indicated limits.Step 1 of 3 : Find limx→2−f(x)lim𝑥→2−𝑓(𝑥).
Consider the piecewise functionf (x) =x + 1, if x < −21, if − 2 ≤ x ≤ 1x2, if x > 1.(i) Find limx→−2 f (x) if it exists
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.