Small Limit Problem in L'Hospital's Rule Chapter

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SUMMARY

The discussion focuses on applying L'Hospital's Rule to solve a limit problem involving the expression \( (e^x + \sin(x))^{1/x} \). The key approach involves taking the natural logarithm of both sides to transform the limit into a suitable form for L'Hospital's Rule. By rewriting the limit as \( \frac{\ln(e^x + \sin(x))}{x} \), both the numerator and denominator approach zero, allowing the application of L'Hospital's Rule. After finding the limit of \( \ln(y) \), the exponential function is used to determine the limit of \( y \).

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Stuck on really starting this one :redface:
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relevant info: Same chapter as L’Hospital’s Rule.

Any ideas?
 
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Since this is in the same chapter as "L'Hopital's rule", it seems likely L'Hopital's rule will work!

To put it into that right form to use for L'Hopital's rule, take the logarithm of both sides:
ln(y)= ln((e^x+ sin(x))^{1/x})= \frac{ln(e^x+ sin(x)}{x}
Use L'Hopital's rule on that (both numerator and denominator go to 0), to find the limit of ln(y) then take the exponential to find the limit of y itself.
 

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