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I am trying to graph this function $c(t)=\frac{2-\sqrt{t+3}}{1-t}$

using this code, to see its behavior at infinity. I need that in the vertical axis appears the value when $t->1$ (The graphs sometimes have 2 cut-off lines when the values are too high, I don't know how to do it) that is 1/4 , and in the axis appears c(t), and if it is possible to give a better representation of the curve.

c[t_]:=\frac{2-\sqrt{t+3}}{1-t}
Plot[c[t], {t, 0, 100000000000}, AxesLabel -> {t, c[t]}]

please help me

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    $\begingroup$ Try this: c[t_] := (2 - Sqrt[t + 3])/(1 - t); LogLogPlot[ c[t], {t, 0, 100000000000}, AxesLabel -> {t, c[t]}] . Have fun! $\endgroup$ Jul 2, 2021 at 10:05
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    $\begingroup$ Try also this: Show[{ Plot[c[t], {t, 0, 2}, AxesLabel -> {Style["t", 16, Italic, Black], Style["c(t)", 16, Italic, Black]}], Graphics[{Gray, Dashed, Line[{{1, 0.235}, {1, 0.270}}], Line[{{0, 0.25}, {2, 0.25}}]}] }]. $\endgroup$ Jul 2, 2021 at 10:16

1 Answer 1

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$Version

(* "12.3.0 for Mac OS X x86 (64-bit) (May 10, 2021)" *)

Clear["Global`*"]

c[t_] = (2 - Sqrt[t + 3])/(1 - t);

Re "to see its behavior at infinity"

Asymptotic was introduced in version 12.1

c2[t_] = Asymptotic[c[t], t -> Infinity]

(* 1/Sqrt[t] *)

Or using Series

c3[t_] = Series[c[t], {t, Infinity, 1}] // Normal

(* -(2/t) + 1/Sqrt[t] *)

LogLogPlot[{c[t], c2[t], c3[t]}, {t, 1, 10^6},
 PlotLegends -> Placed["Expressions", {.4, .35}]]

enter image description here

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