Fitting an exponential function

Question

I want to graph a part of y = e^{\sqrt{x}} for x up to 10^5. I wrote down the following:

\begin{tikzpicture}[scale=0.8, samples=200]
\draw [thick, <->] (0,5) -- (0,0) -- (8,0);
\node [below right] at (7,0) {$x$};
\node [left] at (0,5) {$y$};
\draw[smooth, thick, domain=0:7] plot (\x, {exp((sqrt(\x)))});

However, even though I look for scaling problems over here, somehow I could not find a way to do it. How can I fix the problem?

Answer 1

Here is a starting point, based on your comments. Note that these large numbers may lead to some numerical problems using latex even with floating point calculations.

\documentclass[tikz]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.16}

\begin{document}
\begin{tikzpicture}
\begin{axis}
    \pgfkeys{/pgf/fpu=true}
    \addplot[domain=3400:3600] {exp(sqrt(\x))};
    \addplot coordinates {(3500, 0) (3500,1e26)};
    \pgfkeys{/pgf/fpu=false}
\end{axis}
\end{tikzpicture}
\end{document}

EDIT taking into account additional style demands

\documentclass[tikz]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.16}

\begin{document}
\begin{tikzpicture}
\begin{axis}[xlabel={$x$},ylabel={$y$},xtick=\empty,ytick=\empty,axis x line=bottom, axis y line=left,
    ylabel style={rotate=-90,at={(ticklabel cs:1)}},xlabel style={at={(ticklabel cs:0.875)}}]
    \pgfkeys{/pgf/fpu=true}
    \addplot[color=black,domain=3400:3600] {exp(sqrt(\x))};
    \addplot[color=black] coordinates {(3500, 0) (3500,1e26)};
    \pgfkeys{/pgf/fpu=false}
\end{axis}
\end{tikzpicture}
\end{document}