Friday, 8 August 2014

Find limit limxrightarrow0,fracarctan(3x)tanbig((x+3pi)/3big) without using L'Hospital's rule



Can anybody help me find this limit without using L'Hospital's rule?
limx0arctan(3x)tan((x+3π)/3)
I've tried to multiply both arctan/tan on sin, but it doesn't seem to help.


Answer



Use the following facts.





  • limx0arctan(3x)tan((x+3π)/3)=limx0arctan(3x)3x×3xtanx+3π3


  • tan(π+x3)=tan(x3)



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