Given f left parenthesis x right parenthesis equals open curly brackets table row cell 17 minus 6 x space i f space x less than 3 end cell row cell 2 space i f space x equals 3 end cell row cell 2 minus x space i f space x greater than 3 end cell end table closeUse the given piecewise function to find thelimit as x rightwards arrow 3 of space f left parenthesis x right parenthesis if it exists and also determine if this function is continuous or not when x=3.

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!!!

right

step-by-step explanation: