L'Hopital Rule and limits: [f(2+3x)+f(2+5x)]/x if f(2)=0,...

grapz

Junior Member
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Jan 13, 2007
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I am stuck on this question, can you guys give me some help.

if f' is continuous, f( 2 ) = 0, and f ' (2) = 7, evaluate

lim x --> 0 [ f( 2 + 3x ) + f ( 2 + 5x ) ] / x
 
Re: L'Hopital Rule and limits

Hello, grapz!


Since the limit goes to 00, we can apply L’Hopital’s Rule.\displaystyle \text{Since the limit goes to }\frac{0}{0}\text{, we can apply L'Hopital's Rule.}

limx0[3 ⁣ ⁣f(2 ⁣+ ⁣3x)+5 ⁣ ⁣f(2 ⁣+ ⁣5x)1]    =    3 ⁣ ⁣f(2)+5 ⁣ ⁣f(2)    =    3(7)+5(7)    =    56\displaystyle \lim_{x\to0}\,\left[\frac{3\!\cdot\!f'(2\!+\!3x) + 5\!\cdot\!f'(2\!+\!5x)}{1}\right] \;\;=\;\;3\!\cdot\!f'(2) + 5\!\cdot\!f'(2) \;\;=\;\;3(7) + 5(7) \;\;=\;\;56

 
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