Let,
f(x) = {4, x<5,
-3x, x=5,
10+x, x>5}
Evaluate each of the following:
a) lim(x-->5-)f(x)=
b) lim(x--)5+)f(x)=
c) f(5)=
(a) 4
(b) 15
(c) -15
To evaluate each of the given expressions, we will analyze the function f(x) and its different cases.
a) To find lim(x-->5-)f(x), we need to determine the limit as x approaches 5 from the left side (values less than 5).
From the function definition, we can see that for x < 5, f(x) = 4. So, regardless of approaching 5 from the left side, the function f(x) will always be equal to 4. Hence,
lim(x-->5-)f(x) = 4
b) For lim(x-->5+)f(x), we need to compute the limit as x approaches 5 from the right side (values greater than 5).
From the function definition, we can see that for x > 5, f(x) = 10 + x. So, as x approaches 5 from the right side, f(x) will approach 10 + 5 = 15. Therefore,
lim(x-->5+)f(x) = 15
c) To find f(5), we need to substitute x = 5 into the function definition.
Since x = 5 falls into the case where x = 5, we have f(5) = -3x, which means f(5) = -3 * 5 = -15. Hence,
f(5) = -15
In summary:
a) lim(x-->5-)f(x) = 4
b) lim(x-->5+)f(x) = 15
c) f(5) = -15