Answer
This is not true.
Work Step by Step
Not true. The counterexample is the function
$$f(x)=1+e^{-\frac{1}{x^2}}.$$
This funtion is everywhere greater than $1$ but when $x\to 0$ we have
$$\lim_{x\to0}(1+e^{-\frac{1}{x^2}})=\left[1+e^{-\frac{1}{0^+}}\right]=\left[1+e^{-\infty}\right]=1$$
which is not greater than one.
