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$$\lim_{x\to2^{+}}e^{3/2(2-x)}$$

Properties of the Natural Exponential Function:

The exponential function $f(x)=e^x$ is an increasing continuous function with domain $\mathbb R$ and range $(0, \infty)$. Thus $e^x>0$ for all $x$. Also

$\lim_{x\to-\infty}e^x$=0 and the $\lim_{x\to\infty}$ $e^x=\infty $

So the $x$-axis is a horizontal asymptote of $f(x)= e^x$.

How do I utilize this definition to solve this problem?

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  • $\begingroup$ What is $\lim_{x \to 2^+} 3/(2(2-x))$? $\endgroup$ – Kopper Jul 10 '15 at 15:14
  • $\begingroup$ how must we read $3/2(2-x)$? don't you use $\LaTeX$? $\endgroup$ – Dr. Sonnhard Graubner Jul 10 '15 at 15:18
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Since $$\lim_{x \to 2^{+}} \frac{3}{2(2-x)} = -\infty$$ then $$\lim_{x \to 2^{+}}\exp\left(\frac{3}{2(2-x)}\right) = \exp\left( \lim_{x \to 2^+} \frac{3}{2(2-x)}\right) = 0$$

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  • $\begingroup$ well explained. $\endgroup$ – Madona Syombua Jul 10 '15 at 15:39

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