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Is the function given by ​f(x)equalsleft brace Start 2 By 2 Matrix 1st Row 1st Column one fourth x plus 1 comma 2nd Column for x less than or equals 4 comma 2nd Row 1st Column 4 x minus 11 comma 2nd Column for x greater than 4 comma EndMatrix continuous at xequals4​? Why or why​ not? Choose the correct answer below. A. The given function is continuous at xequals4 because ModifyingBelow lim With x right arrow 4 f (x )does not exist. B. The given function is not continuous at xequals4 because ​f(4​) does not exist. C. The given function is continuous at xequals4 because the limit is 2. D. The given function is not continuous at xequals4 because ModifyingBelow lim With x right arrow 4 f (x )does not exist.

Respuesta :

Answer:

C. The given function is continuous at x=4 because the limit is 2.

Step-by-step explanation:

Given the function:

[tex]f(x)=\left\{\begin{array}{ccc}\frac{1}{4}x+1 &x\leq 4\\4x-11&x>4\end{array}\right[/tex]

We are to determine if the function is continuous at x=4.

For a function to be continuous  at some value c in its domain:

  • f(c) must be defined.
  • [tex]Lim_{x \to c}$ f(x)[/tex] must exist.
  • [tex]f(c)=Lim_{x \to c}$ f(x)[/tex]

Now: at x=4

  • [tex]f(4)=\dfrac{1}{4}*4+1=2[/tex]
  • [tex]Lim_{x \to 4}f(x)=2[/tex]

Since the two values are the same, we say that f(x) is continuous at x=4.

The correct option is C.

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