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Need to calculate the following limit without using L'Hopital's Rule:

$$ \lim_{x \to 0} \frac{5x - e^{2x}+1}{3x +3e^{4x}-3} $$

The problem I'm facing is that no matter what I do I still get expression of the form $$ \frac{0}{0} $$ I thought maybe to use $$ t = e^{2x} $$ But I still can't simplify it enough..

Thank you

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  • $\begingroup$ Is this a homework question? $\endgroup$ – samis Jan 6 '17 at 17:57
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HINT:

As $x\to0,x\ne0$ so safely divide numerator & denominator by $x$

and use

$$\lim_{h\to0}\dfrac{e^h-1}h=1$$

Observe that the exponent of $e,$ the limit variable and the denominator are same.

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  • $\begingroup$ Thank you. Solved it after dividing by x. $\endgroup$ – Noam Jan 6 '17 at 10:19
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$$ \lim_{x \to 0} \frac{5x - e^{2x}+1}{3x +3e^{4x}-3}=\\ \lim_{x \to 0} \frac{5x - (1+2x+\frac{(2x)^2}{2!}+o(x^3))+1}{3x +3(1+4x+\frac{(4x)^2}{2!}+o(x^3))-3} =\\ \lim_{x \to 0} \frac{3x-\frac{(2x)^2}{2!}-o(x^3)}{15x +3\frac{(4x)^2}{2!}+o(x^3)} = \lim_{x \to 0} \frac{3x}{15x } =\frac{3}{15}$$

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    $\begingroup$ $o(x^3)$ is wrong, this is $o(x^2)$ ! Plus no need to go to second order. why complicate things. $\endgroup$ – zwim Jan 6 '17 at 10:10
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Using Taylor's series:

$$\frac{5x - e^{2x}+1}{3x +3e^{4x}-3} = \frac{5x - (1+2x + O(x^2))+1}{3x +3(1+4x+O(x^2))-3} = \frac{3x - O(x^2)}{15x+ O(x^2)} .$$

Thus, $$\lim_{x \rightarrow 0} \frac{3x - O(x^2)}{15x+ O(x^2)} = \frac{1}{5}.$$

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  • $\begingroup$ same remark than for Koshrotash, $o(x^2)$ is wrong. $\endgroup$ – zwim Jan 6 '17 at 10:14
  • $\begingroup$ Why is it wrong? $\endgroup$ – Alex Silva Jan 6 '17 at 10:14
  • $\begingroup$ because $e^x=1+x+x^2/2+o(x^2)$. $\endgroup$ – zwim Jan 6 '17 at 10:15
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    $\begingroup$ @zwim Do you know the difference between $o(x)$ and $O(x)$? Here, $O(x^2)$ is correct. $\endgroup$ – xen Jan 6 '17 at 10:22
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    $\begingroup$ I know they are different techniques; I was saying that your approach "starts to look like" the derivation of L'Hopital. And I am aware that all the answers are (implicitly or not) using the derivative concept. But rather than putting the same comment under every answer, I thought I would vent just once. Don't take it personally - I think it's a problem with the question, not your answer in particular. $\endgroup$ – Floris Jan 6 '17 at 20:16
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Do you know the Taylor formula? If so, then $$ e^x = 1 + x + o(x) $$ as $x \to 0$. Here, $o(x)$ is a function such that $o(x)/x \to 0$ as $x \to 0$. Hence $$ \frac{5x - e^{2x}+1}{3x +3e^{4x}-3} = \frac{5x - (1 + 2x + o(x)) + 1}{3x + 3(1 + 4x + o(x))-3} = \frac{3x+ o(x)}{15x+o(x)} = \frac{3+o(x)/x}{15+o(x)/x} $$ which shows that your limit is $1/5$.

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  • $\begingroup$ $o(x)=xo(1)$, what is this ugly $o(x)/x$. $\endgroup$ – zwim Jan 6 '17 at 10:12
  • $\begingroup$ @zwim This is quite standard notation. I do not see any "ugliness" of $\frac{o(x)}{x}$. $\endgroup$ – xen Jan 6 '17 at 10:24
  • $\begingroup$ I prefer $o(1)$. $\endgroup$ – zwim Jan 6 '17 at 10:25
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Hint: We can divide by $x $ to get $$\lim_{x \to 0} \frac {5-(\frac {e^{2x}-1}{2x}(2))}{3+3 (\frac {e^{4x}-1}{4x}(4))} $$ Can you take it from here?

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    $\begingroup$ Look at lab bhattacharjee's answer. It is exactly what he said. There is a typo in your answer. $\endgroup$ – Behrouz Maleki Jan 6 '17 at 10:09
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$$ \frac{5x - e^{2x}+1}{3x +3e^{4x}-3}=\frac12\frac{\dfrac52-\dfrac{e^{2x}-1}{2x}}{\dfrac34+3\dfrac{e^{4x}-1}{4x}}\to\frac12\frac{\dfrac52-L}{\dfrac34+3L}.$$ (by a scaling of $x$, the two ratios tend to $L$).

Then you can take for granted that $L=1$, giving the answer $\dfrac15$.

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