Question
Evaluate the limit:
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" alt="" />Solution
ATQ, Use the identity: sin(3x) = 3·sin(x) − 4·sin³(x) So, sin(3x) − 3·sin(x) = [3·sin(x) − 4·sin³(x)] − 3·sin(x) = −4·sin³(x) Therefore the limit becomes: lim (x → 0) [ sin(3x) − 3·sin(x) ] / x³ = lim (x → 0) [ −4·sin³(x) / x³ ] = −4 · lim (x → 0) [ (sin(x)/x)³ ] We know: lim (x → 0) sin(x)/x = 1 So: −4 · lim (x → 0) [ (sin(x)/x)³ ] = −4 · 1³ = −4
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