f(x)= 3x+2, x<0
3, x=0
x^2+1, x>0
g(x)= -x+3, x<1
0, x=1
x^2, x>1
find
a) lim x->0+ g(f(x))
b)lim x->0- g(f(x))
c) lim x->1+ f(g(x))
d)lim x->1- f(g(x))
I am trying to understand these. Help appreciated. Have a test soon.
To find the limits in these problems, we need to substitute the given values of x into the functions and see what values they approach as x gets arbitrarily close to the given value.
a) lim x->0+ g(f(x))
To evaluate this limit, we need to find g(f(x)) as x approaches 0 from the right.
Let's begin by finding f(x):
When x < 0, f(x) = 3x + 2.
Since we are looking at the limit as x approaches 0 from the right, we need to find f(0+):
f(0+) = 3(0) + 2 = 2.
Now, let's find g(f(x)):
When x < 1, g(x) = -x + 3.
Since we are looking at the limit of g(f(x)), we substitute f(x) = 2 into g(x):
g(f(x)) = g(2) = -(2) + 3 = 1.
Therefore, lim x->0+ g(f(x)) = 1.
b) lim x->0- g(f(x))
In this case, we need to find g(f(x)) as x approaches 0 from the left.
Let's find f(x) again:
When x < 0, f(x) = 3x + 2.
Since we are looking at the limit as x approaches 0 from the left, we need to find f(0-):
f(0-) = 3(0) + 2 = 2.
Now, let's find g(f(x)):
When x < 1, g(x) = -x + 3.
Substituting f(x) = 2 into g(x):
g(f(x)) = g(2) = -(2) + 3 = 1.
Therefore, lim x->0- g(f(x)) = 1.
c) lim x->1+ f(g(x))
To evaluate this limit, we need to find f(g(x)) as x approaches 1 from the right.
Let's find g(x) first:
When x < 1, g(x) = -x + 3.
Since we are looking at the limit as x approaches 1 from the right, we need to find g(1+):
g(1+) = -(1) + 3 = 2.
Now, let's find f(g(x)):
f(x) = x^2 + 1 for x > 0.
Substituting g(x) = 2 into f(x):
f(g(x)) = f(2) = (2)^2 + 1 = 5.
Therefore, lim x->1+ f(g(x)) = 5.
d) lim x->1- f(g(x))
In this case, we need to find f(g(x)) as x approaches 1 from the left.
Let's find g(x) again:
When x < 1, g(x) = -x + 3.
Since we are looking at the limit as x approaches 1 from the left, we need to find g(1-):
g(1-) = -(1) + 3 = 2.
Now, let's find f(g(x)):
f(x) = x^2 + 1 for x > 0.
Substituting g(x) = 2 into f(x):
f(g(x)) = f(2) = (2)^2 + 1 = 5.
Therefore, lim x->1- f(g(x)) = 5.
I hope this helps you understand how to find limits using function compositions. Good luck on your test!