Don't Overthink It!
What is the value of x → 4 lim f ( x )
If f ( x ) = ⎩ ⎪ ⎨ ⎪ ⎧ ∣ x + 3 − x 2 ∣ 2 1 × ∣ x ∣ + 2 2 0 if − ∞ ≤ x < 2 , and 5 < x ≤ ∞ if 2 ≤ x ≤ 5 for x = 4 if x = 4 .
The answer is 4.
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4 is in the interval of 2 ≤ x ≤ 5 , hence it will fall under f ( x ) = 2 + 2 ∣ x ∣ , but this is not defined at x = 4 as f ( 4 ) = 2 0 . We need to find the limit when x approaches 4, so we have to use two-handed limit.
In this case, 0 < 2 ≤ x ≤ 5 , then x > 0 , so we can remove the absolute value and the two-handed limit will remain the same. We just find the limit of the given expression, which gives: x → 4 lim ( 2 + 2 x ) = 2 + 2 = 4