step by step solution

The given function is

f(x)=\begin{cases}- \frac{x}{2}-\frac{5}{2},\quad{x\lt0}\\\\\ 2x+5 ,\quad\quad\quad{x\geq0}\end{cases}

Notice that as **x** approaches **0** from the left, the function is:

f(x)=- \frac{x}{2}-\frac{5}{2}

To Find the limit:

\lim_{x\to0^{-}}{f(x)}\implies\lim_{x\to0^{-}}\left({ \textcolor{primary}{- \frac{x}{2}-\frac{5}{2}}}\right)

**Direct Substitution Property of Limits**

If **a** is in the domain of a polynomial or a rational function **f**, then:

\lim_{x\to{a}}{f(x)}=f(a)

using the above property we get

\lim_{x\to0^{-}}\left({ \textcolor{primary}{- \frac{x}{2}-\frac{5}{2}}}\right)\implies \left({ {- \frac{\textcolor{primary}{0}}{2}-\frac{5}{2}}}\right)\implies -\frac{5}{2}

The required answer is:

\lim_{x\to0^{-}}{f(x)}=-\frac{5}{2}

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