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Question

$$\underset{x \rightarrow 2^+}{lim}\left(\dfrac{[x]^3}{3} - \left[\dfrac{x}{3}\right]^3\right)$$ is (where [] is g.i.f) is $$\dfrac{a}{3}$$. Find $$a$$.


Solution

$$\lim _{ x\rightarrow { 2 }^{ + } }{ = } \lim _{ h\rightarrow 0 }{ f(2+h) } =\lim _{ h\rightarrow 0 }{ (\cfrac { { [2+h] }^{ 3 } }{ 3 } -{ [\cfrac { 2+h }{ 3 } ] }^{ 3 }) } \\ \Rightarrow (\cfrac { 8 }{ 3 } -0)\\ \Rightarrow \cfrac { 8 }{ 3 } =\cfrac { a }{ 3 } \\ a=8$$

Mathematics

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