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Find limit of 2x+1-cosx/3x as x approaches 0?

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I will assume you meant:
lim (2x + 1 - cosx)/(3x)

then ...

= lim ( (2x)/(3x) + 1/(3x) - (1/3)(cosx/x) ) as x --> 0
= lim ( 2/3 + 0 - (1/3)(0) )
= 2/3

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