complete the definition of the h(x) so that it is continuous over its domain.\nh(x)=\\begin{cases}x^{3},&x <…

complete the definition of the h(x) so that it is continuous over its domain.\nh(x)=\\begin{cases}x^{3},&x < 0\\\\a,&x = 0\\\\\\sqrt{x},&0 < x < 4\\\\b,&x = 4\\\\4-\\frac{1}{2}x,&x > 4\\end{cases}\na = \nb =

complete the definition of the h(x) so that it is continuous over its domain.\nh(x)=\\begin{cases}x^{3},&x < 0\\\\a,&x = 0\\\\\\sqrt{x},&0 < x < 4\\\\b,&x = 4\\\\4-\\frac{1}{2}x,&x > 4\\end{cases}\na = \nb =

Answer

Explanation:

Step1: Find left - hand limit at (x = 0)

(\lim_{x\rightarrow0^{-}}h(x)=\lim_{x\rightarrow0^{-}}x^{3}=0) For the function to be continuous at (x = 0), (\lim_{x\rightarrow0^{-}}h(x)=h(0)), so (a = 0).

Step2: Find left - hand limit at (x = 4)

(\lim_{x\rightarrow4^{-}}h(x)=\lim_{x\rightarrow4^{-}}\sqrt{x}=\sqrt{4}=2)

Step3: Find right - hand limit at (x = 4)

(\lim_{x\rightarrow4^{+}}h(x)=\lim_{x\rightarrow4^{+}}(4-\frac{1}{2}x)=4-\frac{1}{2}\times4=4 - 2=2) For the function to be continuous at (x = 4), (\lim_{x\rightarrow4^{-}}h(x)=h(4)), so (b = 2)

Answer:

(a = 0), (b = 2)