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प्रश्न
Discuss the continuity of the following function at the indicated point:
`f(x) = {{:(|x| cos (1/x)",", x ≠ 0),(0",", x = 0):} at x = 0`
योग
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उत्तर
Given:
`f(x) = {{:(|x| cos (1/x)",", x ≠ 0),(0",", x = 0):} at x = 0`
Consider,
`lim_(x->0) f(x) = lim_(x->0)|x| cos(1/x)`
= `lim_(x->0) f(x) = lim_(x->0) |x| lim_(x->0) cos (1/x)`
= `lim_(x->0) f(x) = 0 xx lim_(x->0) cos (1/x) = 0`
= `lim_(x->0) f(x) = f(0)`
Hence, f(x) is continuous at x = 0
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