Evaluate lim x→1 f(x)
for f(x)= 5x-10,x<1
1 x=1
-3x - 2,x>1
Possible answers:
a) 3
b) -5
c) 1
d) does not exist
for x<1, f(x) = 5x-10, so lim = 5-10 = -5
for x>1, f(x) = -3x-2, so lim = -3-2 = -5
So, the limit is -5, even though f(1) = 1
To evaluate the limit of the function f(x) as x approaches 1, we need to evaluate the values of the function as x approaches 1 from the left and from the right, and check if they converge to the same value.
First, let's evaluate the left-hand limit: lim x→1- f(x). For values of x less than 1, the function is defined as f(x) = 5x - 10. So, we substitute x = 1 into this expression:
lim x→1- f(x) = lim x→1- (5x - 10)
= 5(1) - 10
= -5
Now, let's evaluate the right-hand limit: lim x→1+ f(x). For values of x greater than 1, the function is defined as f(x) = -3x - 2. So, we substitute x = 1 into this expression:
lim x→1+ f(x) = lim x→1+ (-3x - 2)
= -3(1) - 2
= -3 - 2
= -5
Since lim x→1- f(x) = -5 and lim x→1+ f(x) = -5, and they both have the same value, we can say that the limit of f(x) as x approaches 1 exists.
Now, let's evaluate the value of the function at x = 1: f(1).
For x = 1, the function is defined as f(x) = 1. So, we substitute x = 1 into this expression:
f(1) = 1
Finally, we compare the value of f(1) with the limits from the left and right:
lim x→1- f(x) = -5
lim x→1+ f(x) = -5
f(1) = 1
Since the left-hand limit and the right-hand limit do not match with the value of the function at x = 1, we can conclude that the limit of f(x) as x approaches 1 does not exist.
Therefore, the correct answer is: d) does not exist.