Respuesta :
Answer:
a) lim x → 3 f(x) = -1
b) The function is not continuous at the point x=3.
Step-by-step explanation:
f(x) = 17 - 6x if x < 3
f(x) = 2 if x = 3
f(x) = 2 - x if x > 3
Find the limit of f(x) as x → 3
As x tend to 3 from the left hand side
lim x → 3⁻ f(x) = lim x → 3⁻ [17 - 6x]
= 17 - 6(3) = -1
As x tend to 3 from the right hand side
lim x → 3⁺ f(x) = lim x → 3⁺ [2 - x]
= 2 - 3 = -1
The conditions for continuity of a function at point (a), include:
- The function must exist at that point. That is, f(a) is finite.
- The limit of the function as x→a must exist (i.e. the right-hand limit = left-hand limit, and both are finite).
- The limit of the function as x → a must be equal to f(a).
Mathematically,
lim x→a⁻ f(x) = lim x→a⁺ f(x) = lim x→a f(x) = f(a)
This can be interpreted as there being no jump(s) in the graph of the function.
For our question,
lim x → 3⁻ f(x) = -1
lim x → 3⁺ f(x) = -1
lim x → 3 f(x) = -1
f(3) = 2
For the function to be continuous,
lim x→ 3⁻ f(x) = lim x → 3⁺ f(x) = lim x → 3 f(x) = f(3)
But,
-1 = -1 = -1 ≠ 2
Hence, the function is not continuous at the point x=3.
Hope this Helps!!!
