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Question
Let g(x, y) = `("e"^y sin x)/x` for x ≠ 0 and g(0, 0) = 1 shoe that g is continuous at (0, 0)
Solution
g(x, y) = `("e"^y sin x)/x` for x ≠ 0 and g(0, 0) = 1
By definition g(0, 0) = 1 ......(Given)
(i.e.,) `lim_((x, y) -> (0, 0)) "g"(x, y) = lim_((x, y) -> (0, 0)) ("e"^y sin x)/x`
= `[lim_((x, y) -> (0, 0))] [lim_((x, y) -> (0, 0)) sin x]`
= (1)(1)
= 1
Now, limit along the line y = mx
Passing through (0, 0)
`lim_(x, mx) ("e"^(mx) sinx)/x` = 1 ......(Defined)
Hence, 'g' is continuous at (0, 0)
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