Questions: Use the given formal definition to prove the limit statement. lim x -> 0^- (1/x) = -∞ The function f(x) approaches minus infinity as x approaches c from the left, written lim x -> c^- f(x) = -∞, if, for every negative real number -B there exists a corresponding number δ>0 such that for all x, c-δ<x<c -> f(x)<-B. Complete the introduction statement. Given -B<0, find δ>0 such that for all x, <x< -> 1/x<-B.

Use the given formal definition to prove the limit statement.

lim x -> 0^- (1/x) = -∞

The function f(x) approaches minus infinity as x approaches c from the left, written lim x -> c^- f(x) = -∞, if, for every negative real number -B there exists a corresponding number δ>0 such that for all x, c-δ<x<c -> f(x)<-B.

Complete the introduction statement.
Given -B<0, find δ>0 such that for all x, <x< -> 1/x<-B.

Transcript text: Use the given formal definition to prove the limit statement. \[ \lim _{x \rightarrow 0^{-}} \frac{1}{x}=-\infty \] The function $f(x)$ approaches minus infinity as $x$ approaches c from the left, written $\lim _{x \rightarrow c^{-}} f(x)=-\infty$, if, for every negative real number $-B$ there exists a corresponding number $\delta>0$ such that for all $x, c-\delta0$ such that for all $\mathrm{x}, \square<\mathrm{x}<\square \rightarrow \frac{1}{\mathrm{x}}<-\mathrm{B}$.

Solution

Solution Steps

To prove the limit statement using the formal definition, we need to show that for every negative real number $-B$, there exists a $\delta > 0$ such that for all $x$ in the interval $(c-\delta, c)$, the inequality $f(x) < -B$ holds. In this case, $f(x) = \frac{1}{x}$ and $c = 0$. We need to find $\delta$ such that for all $x$ in the interval $(- \delta, 0)$, $\frac{1}{x} < -B$.

Solution Approach

Start by considering the inequality $\frac{1}{x} < -B$.
Solve for $x$ to find the condition that $x$ must satisfy.
Determine $\delta$ based on this condition.

Step 1: Understand the Limit Definition

To prove $\lim_{x \to 0^-} \frac{1}{x} = -\infty$, we need to show that for every negative real number $-B$, there exists a $\delta > 0$ such that for all $x$ in the interval $(- \delta, 0)$, the inequality $\frac{1}{x} < -B$ holds.

Step 2: Solve the Inequality

Consider the inequality $\frac{1}{x} < -B$. Solving for $x$, we get: \[ x < -\frac{1}{B} \] This means that for the inequality to hold, $x$ must be less than $-\frac{1}{B}$.

Step 3: Determine $\delta$

Since we need $x$ to be in the interval $(- \delta, 0)$, we choose $\delta$ such that: \[ \delta = -\frac{1}{B} \] This ensures that for all $x$ in $(- \delta, 0)$, the inequality $\frac{1}{x} < -B$ is satisfied.