From Donaldson's Riemann Surfaces- Existence of non-vanishing $C^\infty$ function $ f(t)= 1$ for small $t$, $f(t)=\frac{1}{t}$for large $t$ Does there exist a $C^\infty$ function $p:\mathbb{R} \to \mathbb{R}$ such that  p does not vanish and
$p(t)=1$ for small $t$ and $p(t)= \frac{1}{t}$ for large $t$? 
Such function $p$ is used in Donaldson's book Riemann Surfaces p.100 Proposition 18. but I can't find such $p$.

Proposition 18. Let $\alpha$ be a non-trivial meromorphic 1-form on a compact Riemann surface $X$ with genus $g$. Then the number of zeros minus number of poles of $\alpha $, counting according to multiplicity, is $2g-2$
Proof. Fix an area form $\omega$ on $X$. 
Let $X'$ be the set on which $\alpha$ is holomorphic.
Define a Hermitian metric on $T^*X'$ as $ \vert \alpha \vert ^2 \omega = \alpha \wedge \bar{\alpha}$. 
Choose a real-valued functoin $p$ on $\mathbb{R}$ with $p(t)=1$ for small to and $p(t)= \frac{1}{t}$ for large t.
Define $\widetilde{\alpha} := \begin{cases}
      p(\vert \alpha \vert ^2)\cdot \alpha
       &\text { on $X'$}
    \\[4pt]
     0
       &\text{ on $X-X'$}
    \end{cases} $
Let $q$ be a pole of $\alpha$, $z$ be a coordinate fucntion centered at $q$, $\alpha = f(z) dz$  near $q$. 
Near $q$, $\omega = R \cdot dz\wedge d\bar{z}$ for some strictly positive function $R$.
$\widetilde{\alpha} = \frac{1}{\vert \alpha \vert ^2} f \ dz=\frac{R}{\vert f \vert ^2} f \ dz = \frac{R}{\bar{f}}  \ dz$.
Thus $\widetilde{\alpha} $ is smooth 1-form on $X$ and its zero set is the union on zeros and poles of $\alpha$.

Question. For $\alpha$ to be smooth, $p$ has to be a smooth function and for the zero set of $\widetilde{\alpha}$ to be the union on zeros and poles of $\alpha$, $p$ should not vanish. 
Does there exist a $C^\infty$ function $p:\mathbb{R} \to \mathbb{R}$ such that  p does not vanish and
$p(t)=1$ for small $t$ and $p(t)= \frac{1}{t}$ for large $t$? 
I know that $(1-g(t)) \cdot k(t)$ is a smooth function with $k(t)=1$ for small 
 $t$ and $k(t)= \frac{1}{t}$ for large  $t$ ,  but $k$ vanishes on $(2/3 , \ 4/3)$
$ g(t) := \frac{\int_{|t-1|}^{2/3} h(u) du }{\int_{1/3}^{2/3} h(u) du}$ is a bump function equal to 1 on $[2/3 , \ 4/3]$ supported in $[1/3, \ 5/3]$
where $h(u)  := \begin{cases}
       exp\{-(u-1/3)^{-1} (u-2/3)^{-1}\}
       &\text { on $[1/3,\ 2/3]$}
    \\[4pt]
     0
       &\text{ otherwise}
    \end{cases} $,
$k(t):= \begin{cases}
       1
       &\text{ on $(-\infty,\ 1]$}
    \\[4pt]
     \frac{1}{t}
       &\text{ otherwise}
    \end{cases}$

Take the function $f(t):=1$ $(t\leq 1)$ and $:={1\over t}$ $(t\geq1)$, and convolve it with a $C^\infty$ bump function $\phi$ having support in $\bigl[-{1\over2},{1\over2}\bigr]$:
$$g(x):=\int_{-\infty}^\infty f(x-t)\>\phi(t)\>dt\ .$$
Here it is assumed that $\phi(t)\geq0$ and $\int_{-\infty}^\infty \phi(t)\>dt=1$.
