Find the limit L. Then find delta > 0 such that |f(x) - L| < 0.01 whenever 0 < |x-c| < delta.
lim x -> 4 (4- (x/2))
To find the limit of the function f(x) as x approaches 4, you can simply substitute the value 4 into the function and evaluate the expression:
f(4) = 4 - (4/2) = 4 - 2 = 2
So, the limit of the function f(x) as x approaches 4 is 2.
To find delta > 0 such that |f(x) - L| < 0.01 whenever 0 < |x-c| < delta, you need to find a suitable value of delta that ensures the difference between f(x) and the limit L is less than 0.01 when the distance between x and c is less than delta.
In this case, we have L = 2 and c = 4. Let's start by setting |f(x) - L| < 0.01:
|f(x) - 2| < 0.01
Now let's work with the absolute value expression:
|4 - (x/2) - 2| < 0.01
Simplifying the expression inside the absolute value:
|2 - (x/2)| < 0.01
To find a suitable value of delta, we need to bound the expression inside the absolute value:
-0.01 < 2 - (x/2) < 0.01
Next, solve for x:
2 - 0.01 < (x/2) < 2 + 0.01
1.99 < (x/2) < 2.01
Multiplying both sides by 2:
3.98 < x < 4.02
From the inequality, we can deduce that if x lies within the interval (3.98, 4.02), the difference between f(x) and 2 will be less than 0.01. So, we can choose delta = 0.02 as our suitable value.
Therefore, if 0 < |x - 4| < 0.02, then |f(x) - 2| < 0.01.