Test the continuity of the function f(x) = |x| for x = 0.

1.	Test the continuity of the function f(x) = \|x\| for x = 0.
Answer» Given function f(x) = \|x\| = x, x > 0 = -x, x < 0 = 0, x = 0 Hence f(0) = 0 Now, Left Hand Limit = lim_{x → 0}- f(x) = lim_{x → 0} (-x) = 0 [∵ f(x) = -x, x < 0] Right Hand Limit = lim_{x → 0}+ f(x) = lim_{x → 0} (x) = 0 So, L.H.L. = R.H.L. = f(0) Hence, f(x) is continuous at x = 0 [This can also be verified by graph of \|x\|]

Answer»

Given function f(x) = |x|

= x, x > 0 = -x, x < 0 = 0, x = 0

Hence f(0) = 0

Now, Left Hand Limit = lim_{x → 0}- f(x) = lim_{x → 0} (-x) = 0 [∵ f(x) = -x, x < 0]

Right Hand Limit = lim_{x → 0}+ f(x) = lim_{x → 0} (x) = 0

So, L.H.L. = R.H.L. = f(0)

Hence, f(x) is continuous at x = 0

[This can also be verified by graph of |x|]

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