Find Limit in 2 Mins - Tricks & Tips

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SUMMARY

The discussion focuses on the limit calculation of the expression $$\lim_{x\to\infty} x(e^{\frac 1x}-1)$$, which simplifies to the derivative of the exponential function at zero. The limit is evaluated using the transformation to $$\lim_{x\to 0} \frac{1}{x}(e^x-1)$$, ultimately confirming that the result is 1, as it represents the derivative of $$e^x$$ at $$x=0$$. This method highlights the application of derivatives in limit evaluations.

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  • Understanding of limits in calculus
  • Familiarity with derivatives and their properties
  • Knowledge of exponential functions, specifically $$e^x$$
  • Basic algebraic manipulation skills
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  • Explore the rules and applications of derivatives
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  • Practice limit evaluations using L'Hôpital's Rule
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Students of calculus, mathematics educators, and anyone interested in advanced limit evaluation techniques.

Lorena_Santoro
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We might recognize the shape of a derivative.
$$\lim_{x\to\infty} x(e^{\frac 1x}-1)
=\lim_{x\to 0} \frac 1x(e^x-1)
=\lim_{x\to 0} \frac{e^x-e^0}{x-0}$$

This is the derivative of $e^x$ at $x=0$, which is $e^0=1$.
 
Nice one!
 

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