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  • Is the function f defined by f x = x if x < 1 5 if x > 1 continuous at x = 0 ? At x =1? At x =2?

5. Is the function f defined by  
f (x) = \left\{\begin{matrix} x , & if x \leq 1 \\ 5 & if x \geq 1 \end{matrix}\right.
continuous at x = 0? At x = 1? At x = 2?

Answers (1)

best_answer

Given function is
f (x) = \left\{\begin{matrix} x , & if x \leq 1 \\ 5 & if x \geq 1 \end{matrix}\right.
function is defined at x = 0 and its value is 0
f(0) = 0\\ \lim_{x\rightarrow 0}f(x) = f(x) = 0\\ \lim_{x\rightarrow 0}f(x) = f(0)
Hence , given function is continous at x = 0

given function  is defined for x = 1
Now, for x = 1 Right-hand limit and left-hand limit are not equal
f(1) = 1\\ \lim_{x\rightarrow 1^-}f(x) = f(x) = 1\\ \lim_{x\rightarrow 1^+}f(x) =f(5) = 5
R.H.L \neq L.H.L.
Therefore, given function is not continous at x =1
Given function is defined for x = 2 and its value at x = 2 is 5
f(2) = 2\\ \lim_{x\rightarrow 2}f(x) = f(5) = 5\\\lim_{x\rightarrow 2}f(x) = f(2)

Hence, given function is continous at x = 2

Posted by

Gautam harsolia

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