Question

We write \(\lim _{x \rightarrow a} f(x)=-\infty\) if for any negative number \(M\) there exists \(a \delta>0\) such that $$f(x) < M \quad \text { whenever } \quad 0< |x-a| < \delta$$ Use this definition to prove the following statements. $$\lim _{x \rightarrow-2} \frac{-10}{(x+2)^{4}}=-\infty$$

Short answer

Question: Prove that the limit of the function \(f(x) = \frac{-10}{(x+2)^{4}}\) as \(x\) approaches \(-2\) is \(-\infty\). Answer: By using the definition of the limit approaching \(-\infty\), we showed that for any negative number \(M\), there is a positive number \(\delta\) such that \(f(x) < M\) whenever \(0 < |x + 2| < \delta\). Our choice of \(\delta\) was \(\sqrt[4]{\frac{-10}{M}}\). Thus, we have proved that $$\lim _{x \rightarrow -2} \frac{-10}{(x+2)^{4}}=-\infty$$

Step-by-step solution

Define the Function

We are given the function \(f(x) = \frac{-10}{(x+2)^{4}}\). We want to prove that \(\lim_{x\rightarrow -2}f(x) = -\infty\) using the definition provided.

Given Definition of Limit

The definition states that for any negative number \(M\), there exists \(\delta>0\) such that $$f(x)<M \quad \text { whenever } \quad 0<|x+2|<\delta$$

Use Given Information

The function \(f(x) = \frac{-10}{(x+2)^{4}}\). In order to show that \(f(x) < M\) for any negative number \(M\), we need to find a relationship between \(M\) and \(\delta\). Substitute the function into the inequality: $$\frac{-10}{(x+2)^{4}} < M$$ Now, we need to find a value for \(\delta>0\) such that this inequality holds for all negative \(M\).

Manipulate the Inequality

First, let's multiply both sides of the inequality by \((x + 2)^4\) so we can isolate \(M\). Since \((x + 2)^4 > 0\) for all \(x\), the inequality remains the same: $$-10 < M(x+2)^{4}$$ Now, divide both sides by \(M\): $$\frac{-10}{M} < (x+2)^{4}$$ Then, apply the \(4\)-th root to both sides: $$\sqrt[4]{\frac{-10}{M}} < x+2$$ Since \(M\) is negative and \(x\) approaches \(-2\), the inequality becomes $$0 < |x + 2| < \sqrt[4]{\frac{-10}{M}}$$

Define \(\delta\) and Prove the Statement

From the inequality, we can choose \(\delta = \sqrt[4]{\frac{-10}{M}}\). This will satisfy the inequality for any negative number \(M\). Thus, we have shown that for any negative number \(M\), there exists a \(\delta\) such that \(f(x) < M\) whenever \(0 < |x + 2| < \delta\). Therefore, we have proved that $$\lim _{x \rightarrow -2} \frac{-10}{(x+2)^{4}}=-\infty$$

Key concepts

Infinity in Limits

Infinity in limits deals with the behavior of functions when the input approaches a point but the function's output becomes infinitely large or small. This is often expressed with limits that approach positive or negative infinity. For example, when we say \( \lim_{x\rightarrow a} f(x) = -\infty \), it means as \( x \) gets close to \( a \), \( f(x) \) decreases without bound, approaching negative infinity. In the provided exercise, the function is \( f(x) = \frac{-10}{(x+2)^{4}} \) and we are interested in the behavior as \( x \) approaches \(-2\). Here, \((x+2)\) will approach zero causing \(\frac{1}{(x+2)^{4}}\) to approach infinity, making the entire expression approach \(-\infty\) because of the negative sign. The function decreases sharply, demonstrating the concept of limits involving infinity.

Delta-Epsilon Definition

The delta-epsilon definition provides a rigorous way to define the limit of a function. It involves two parameters: "delta" (\(\delta\)) and "epsilon" (\(\epsilon\)). For a limit \( \lim_{x \to a} f(x) = L \), the formal definition states: for every positive number \(\epsilon\), there exists a positive number \(\delta\) such that if \(0 < |x-a| < \delta\) then \(|f(x) - L| < \epsilon\).

In the context of going to infinity, the same principle applies but focuses on the function being larger or smaller than a number \(M\), often involving infinity. For negative limits like \( \lim_{x \rightarrow a} f(x) = -\infty \), the definition adjusts to ensure \(f(x) < M\) whenever \(0< |x-a| < \delta\), for any negative \(M\). This ensures that as \(x\) approaches \(a\), the function decreases beyond any negative bound \(M\). In this exercise, we identified such a \(\delta\) that maintains the condition, connecting delta-epsilon with infinity limits.

Manipulating Inequalities

Manipulating inequalities is crucial for solving limits and understanding their behavior. The process often involves algebraic manipulations to isolate variables or expressions to show a relationship. In our exercise, we needed to show \( f(x) < M \), where \( f(x) = \frac{-10}{(x+2)^{4}} \), for any negative number \(M\).

To achieve this, you follow these steps:

• Multiply through to eliminate fractions: Start by multiplying both sides by \((x+2)^4\) which maintains the inequality direction because \((x+2)^4\) is positive.
• Divide to isolate the expression: Next, divide by \(M\) to get \(\frac{-10}{M} < (x+2)^4\).
• Solve using roots: Apply the fourth root to both sides to solve for \(x+2\), leading to the expression \(\sqrt[4]{\frac{-10}{M}} < |x+2|\).

Finally, setting \(\delta = \sqrt[4]{\frac{-10}{M}}\) ensures that every negative \(M\) leads to the desired condition being satisfied, demonstrating successful manipulation of the inequality to fulfill the limit requirements.