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### Limit of $\lim\limits_{x \to -3} \frac {4^\frac{x+3}{5}-1}{x+3}$, without L'Hopital's rule.

HINT Are you acquainted to the derivative definition? \begin{align*} \lim_{x\to-3}\frac{4^{\frac{x+3}{5}} - 1}{x + 3} = \lim_{x\to-3}\frac{4^{\frac{x+3}{5}} - 4^{\frac{-3 + 3}{5}}}{x - (-3)} \end{...

### Show that $(1 + x^2 y^2)^{-\frac{1}{x^2 + y^2}}$ is $1$ as $(x,y) \to(0,0)$

Hint Take the natural logarithm and use that $z-\frac{z^2}2\le \ln(1+z) \le z$. Additionally we can switch to radial coordinates $x=r\cos\theta,\, y=r\sin\theta$ and let $r\to 0$.
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### Calculate $\lim_{x \to 0} \frac{\cos(x)}{\sin(x)}$

I would give you full score without any hesitation. Some professors would ask for an $\epsilon$-$\delta$ proof but in my opinion this would be counter productive. Here are some reasons why I like your ...
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### Limit of $\lim\limits_{x \to -3} \frac {4^\frac{x+3}{5}-1}{x+3}$, without L'Hopital's rule.

If you prefer a proof without using derivatives: Observe that our limit is also equal to $$\lim\limits_{x\to-3}\frac {4^{(x+3)/5}-1}{x+3}=\frac 15\lim\limits_{x\to0}\frac {4^x-1}x$$ Where the ...
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### Show that $(1 + x^2 y^2)^{-\frac{1}{x^2 + y^2}}$ is $1$ as $(x,y) \to(0,0)$

You can also observe that  \begin{array}{l} \left( {1 + x^2 y^2 } \right)^{ - \frac{1}{{x^2 + y^2 }}} = e^{ - \frac{1}{{x^2 + y^2 }}\log \left( {1 + x^2 y^2 } \right)} = \\ \\ = e^{ - \...

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