Let $h(x)=-\frac{5}{x}$ . $\newline$ Select the correct description of the one-sided limits of $h$ at $x=0$ . $\newline$ Choose $1$ answer: $\newline$ (A) $\lim _{x \rightarrow 0^{+}} h(x)=+\infty$ and $\lim _{x \rightarrow 0^{-}} h(x)=+\infty$ $\newline$ (B) $\lim _{x \rightarrow 0^{+}} h(x)=+\infty$ and $\lim _{x \rightarrow 0^{-}} h(x)=-\infty$ $\newline$ (C) $\lim _{x \rightarrow 0^{+}} h(x)=-\infty$ and $\lim _{x \rightarrow 0^{-}} h(x)=+\infty$ $\newline$ (D) $\lim _{x \rightarrow 0^{+}} h(x)=-\infty$ and $\lim _{x \rightarrow 0^{-}} h(x)=-\infty$

Full solution

Q. Let

h(x)=-\frac{5}{x}

\newline

Select the correct description of the one-sided limits of

h

x=0

\newline

Choose

1

answer:

\newline

(A)

\lim _{x \rightarrow 0^{+}} h(x)=+\infty

and

\lim _{x \rightarrow 0^{-}} h(x)=+\infty

\newline

(B)

\lim _{x \rightarrow 0^{+}} h(x)=+\infty

and

\lim _{x \rightarrow 0^{-}} h(x)=-\infty

\newline

(C)

\lim _{x \rightarrow 0^{+}} h(x)=-\infty

and

\lim _{x \rightarrow 0^{-}} h(x)=+\infty

\newline

(D)

\lim _{x \rightarrow 0^{+}} h(x)=-\infty

and

\lim _{x \rightarrow 0^{-}} h(x)=-\infty

Approaching $0$ from the positive side: Consider the function $h(x) = -\frac{5}{x}$ and what happens as $x$ approaches $0$ from the positive side (right-hand limit). As $x$ gets closer to $0$ from the positive side, the value of $\frac{1}{x}$ becomes very large, and since we have a negative sign in front of the $5$ , $h(x)$ will approach negative infinity.
Calculating the right-hand limit: Calculate the right-hand limit of $h(x)$ as $x$ approaches $0$ . $\lim_{x \rightarrow 0^{+}}h(x) = \lim_{x \rightarrow 0^{+}}-\frac{5}{x} = -\infty$
Approaching $0$ from the negative side: Consider the function $h(x) = -\frac{5}{x}$ and what happens as $x$ approaches $0$ from the negative side (left-hand limit). As $x$ gets closer to $0$ from the negative side, the value of $\frac{1}{x}$ becomes very large in the negative direction, and since we have a negative sign in front of the $5$ , $h(x)$ will approach positive infinity.
Calculating the left-hand limit: Calculate the left-hand limit of $h(x)$ as $x$ approaches $0$ . $\lim_{x \to 0^{-}}h(x) = \lim_{x \to 0^{-}}-\frac{5}{x} = +\infty$