<s>[INST] Limit of a complicated function. Find $$\lim\limits_{x \to 2^{-}} \frac{e^{((x+2)\log 4){\frac{[x+1]}{4}}}-16}{ 4^x -16}$$ where $[x]$ denotes the greatest integer function less than or equal to x.
ATTEMPT:
I tried the following substitution $x=2-h$ and as $x \to 2$, $h \to 0.$ which gave me something like this
$\lim\limits_{h\to 0}$$ e^{((4-h)ln4){\frac{[3-h]}{4}}}-16\over \frac{16}{4^h} -16.$The denominator seems solvable as we get $\lim\limits_{h\to 0}$ $-16\frac{(4^h-1)h}{4^h*h}$ which seems a standard limit.But the numerator stills remain the same.
How can i simplify the numerator?
 [/INST] If we assume that there is a typo in the question (as mentioned in my comment) then (The mentioned typo has been fixed by OP)
We can write the given function as $$f(x) = \dfrac{\exp\left\{\{(x + 2)\log 4\}\cdot\dfrac{[x + 1]}{4}\right\} - 16}{4^{x} - 16}$$ When $x \to 2^{-}$ we can write $[x + 1] = 2$ and then the expression for $f(x)$ can be simplified to $$f(x) = \dfrac{\exp\left\{\left(1 + \dfrac{x}{2}\right)\log 4\right\} - \exp(2\log 4)}{4^{x} - 16}$$ Putting $x = 2 - h$ where $h \to 0^{+}$ we can see that the desired limit $L$ can be calculated as follows
\begin{align}
L &= \lim_{h \to 0^{+}}\dfrac{\exp\left\{\left(2 - \dfrac{h}{2}\right)\log 4\right\} - \exp(2\log 4)}{4^{2 - h} - 16}\notag\\
&= \exp(2\log 4)\lim_{h \to 0^{+}}\dfrac{\exp\left\{\left(2 - \dfrac{h}{2}\right)\log 4 - 2\log 4\right\} - 1}{\dfrac{16}{4^{h}} - 16}\notag\\
&= \frac{\exp(2\log 4)}{16}\lim_{h \to 0^{+}}4^{h}\cdot\dfrac{\exp\left\{\left(2 - \dfrac{h}{2}\right)\log 4 - 2\log 4\right\} - 1}{1 - 4^{h}}\notag\\
&= 1\cdot\lim_{h \to 0^{+}}1\cdot\dfrac{\exp\left\{\left(2 - \dfrac{h}{2}\right)\log 4 - 2\log 4\right\} - 1}{h}\cdot\frac{h}{1 - 4^{h}}\notag\\
&= -\frac{1}{\log 4}\lim_{h \to 0^{+}}1\cdot\dfrac{\exp\left\{\left(2 - \dfrac{h}{2}\right)\log 4 - 2\log 4\right\} - 1}{h}\notag\\
&= -\frac{1}{\log 4}\lim_{h \to 0^{+}}1\cdot\dfrac{\exp\left\{\left(2 - \dfrac{h}{2}\right)\log 4 - 2\log 4\right\} - 1}{\left(2 - \dfrac{h}{2}\right)\log 4 - 2\log 4}\cdot\dfrac{\left(2 - \dfrac{h}{2}\right)\log 4 - 2\log 4}{h}\notag\\
&= -\frac{1}{\log 4}\lim_{t \to 0^{-}}\frac{e^{t} - 1}{t}\cdot\left(-\frac{\log 4}{2}\right)\notag\\
&= \frac{1}{2}
\end{align}
In the above derivation we have used substitution $$t = \left(2 - \dfrac{h}{2}\right)\log 4 - 2\log 4$$ and $t \to 0^{-}$ when $h \to 0^{+}$. Also the following standard limits are used $$\lim_{x \to 0}\frac{e^{x} - 1}{x} = 1, \lim_{x \to 0}\frac{a^{x} - 1}{x} = \log a$$
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