Which rule or method can be applied against lim x → 0 1...

ntaraxq

ntaraxq

Answered

2022-07-07

Which rule or method can be applied against   lim x 0 1 + 6 x ( 1 + 3 x ) x 2   ? ?\

Answer & Explanation

Karla Hull

Karla Hull

Expert

2022-07-08Added 20 answers

L = lim x 0 1 + 6 x ( 1 + 3 x ) x 2
= lim x 0 ( 1 + 6 x ( 1 + 3 x ) ) x 2 ( 1 + 6 x + ( 1 + 3 x ) ) ( 1 + 6 x + ( 1 + 3 x ) )
= lim x 0 ( 1 + 6 x ) ( 1 + 3 x ) 2 x 2 ( 1 + 6 x + ( 1 + 3 x ) )
= lim x 0 1 + 6 x ( 1 + 6 x + 9 x 2 ) x 2 ( 1 + 6 x + ( 1 + 3 x ) )
= lim x 0 1 + 6 x 1 6 x 9 x 2 x 2 ( 1 + 3 x + 1 + 6 x )
= lim x 0 9 x 2 x 2 ( 1 + 3 x + 1 + 6 x )
= lim x 0 9 ( 1 + 3 x + 1 + 6 x )
= lim x 0 9 ( 1 + 1 ) = 9 2
auto23652im

auto23652im

Expert

2022-07-09Added 5 answers

From your line 4:
lim x 0 3 1 + 6 x 3 2 x 0 0
So by L'Hopital's Rule,
lim x 0 3 1 + 6 x 3 2 x = lim x 0 3 2 ( 1 + 6 x ) 3 / 2 6 2
= 3 2 ( 1 + 6 ( 0 ) ) 3 / 2 6 2 = 9 2

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