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Question
Let g(x, y) = `(x^2y)/(x^4 + y^2)` for (x, y) ≠ (0, 0) = 0. Show that `lim_((x, y) -> (0, 0)) "g"(x, y)` = 0 along every line y = mx, m ∈ R
Solution
Given g(x, y) `(x^2y)/(x^4 + y^2)` for (x, y) ≠ (0, 0) and f(0, 0) = 0
Along the line y = mx, m ∈ R
`lim_((x, y) -> (0, 0)) "g"(x, y) = lim_((x, mx) -> (0, 0)) ((x^2("m"x))/(x^4 + ("m"x)^2))`
= `lim_((x, mx) -> (0, 0)) (("m"x^3)/(x^4 + "m"^2x^2))`
= `lim_((x, mx) -> (0, 0)) (("m"x)/(x^2 + "m"^2))`
= `0/"m"^2`
= 0
Hence proved.
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