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. $$\lim _{x \rightarrow 1} \frac{-2}{(x-1)^{2}}=-\infty$$

Short answer

Question: Prove that the limit of the function \(f(x)=\frac{-2}{(x-1)^{2}}\) is \(-\infty\) as \(x\) approaches \(1\). Answer: To prove that the limit is \(-\infty\), we showed that for any negative number \(M\), there exists a positive number \(\delta\) such that when \(0<|x-1|<\delta\), the function \(f(x)<M\). We found the value of \(\delta\) in terms of \(M\) as \(\delta=\sqrt{\frac{1}{|-M|/2}}\), and showed that this satisfies the given definition of the limit. Therefore, we can conclude that \(\lim _{x \rightarrow 1} \frac{-2}{(x-1)^{2}}=-\infty\).

Step-by-step solution

Recall the definition of the limit

Recall the definition of the limit where \(\lim _{x \rightarrow a} f(x)=-\infty\). This means that for any negative number \(M\), there exists a positive number \(\delta\) such that \(f(x) < M\) whenever \(0< |x-a| < \delta\). In this problem, we have \(f(x) = \frac{-2}{(x-1)^{2}}\) and \(a = 1\). We are asked to prove that \(\lim _{x \rightarrow 1} f(x) = -\infty\) using the given definition.

Manipulate the inequality involving f(x)

We need to show that for any \(M<0\), there is a positive \(\delta\) such that \(0< |x-1| < \delta\) implies that \(f(x) < M\). We can rewrite the inequality involving \(f(x)\) as the following inequality to find \(\delta\) in terms of \(M\): $$\frac{-2}{(x-1)^{2}} < M$$

Find \(\delta\) in terms of M

Now, we solve the inequality in terms of \(\delta\): $$\frac{1}{|-M|/2}<(x-1)^2$$ $$\sqrt{\frac{1}{|-M|/2}}<|x-1|$$ Let \(\delta=\sqrt{\frac{1}{|-M|/2}}\). Then, if \(0<|x-1|<\delta\), we have: $$0<|x-1|<\sqrt{\frac{1}{|-M|/2}}$$ Which implies: $$f(x)=\frac{-2}{(x-1)^{2}}<M$$

Conclusion

We have shown that for any negative number \(M\), there exists a positive number \(\delta\) such that when \(0<|x-1|<\delta\), the function \(f(x)=\frac{-2}{(x-1)^{2}}\) is less than \(M\). Therefore, by definition, we can conclude that: $$\lim _{x \rightarrow 1} \frac{-2}{(x-1)^{2}}=-\infty$$

Key concepts

Epsilon-Delta Definition

The epsilon-delta definition is a fundamental way of understanding the limits of functions. It is predominantly used to prove that a function approaches a certain value as the input approaches a specific point. For limits approaching infinity, a similar concept applies, particularly in proving a function approaches negative infinity.
In such cases, for any negative number, denoted by \(M\), there exists a small positive value, represented as \(\delta\), where the function satisfies certain conditions within a specified range. This implies that the function's values become less than any predefined number \(M\), for inputs close enough to the target point, but not exactly equal to it.
The core idea is to determine how the function behaves by connecting closeness in input to behavior in output, hence creating a mathematical assurance of the function moving towards a certain trend or value, in this case, negative infinity.

Limits Approaching Infinity

When we talk about limits approaching infinity, it involves understanding how a function behaves as the input grows larger and larger. However, the context can also include how a function creeps towards negative infinity as the input reaches a specific point.
This is particularly useful in scenarios where a function like \(f(x) = \frac{-2}{(x-1)^2}\) is analyzed as \(x\) nears 1. Here, the function's output dramatically decreases without bounds, signifying a descent into negative infinity.

• Consider expressions or situations driving towards infinite growth or reduction in output.
• The essential task is to calculate that every decrease or increase fulfills the condition provided for the specific range of inputs close to the point of interest.

Limits approaching infinity help us determine behaviors at boundaries, leading to profound understandings and proofs in mathematical calculus.

Negative Infinity Limit

Understanding negative infinity limits involves studying situations where a function's output becomes arbitrarily large in the negative direction as the input approaches a precise value. Such understanding often requires demonstrating this behavior formally, as is done with the function \(f(x) = \frac{-2}{(x-1)^2}\) near \(x = 1\).
The crucial steps involve:

• Recognizing that as the denominator \((x-1)^2\) becomes very small, \(f(x)\) tends sharply towards negative infinity due to the negative numerator.
• Formulating inequalities and deriving \(\delta\) that validate the output for values smaller than a selected negative \(M\).
• Concluding that conditions are met, thereby proving the function decreases endlessly.

Essentially, a negative infinity limit makes explicit the rapid descent of function values, forming a cornerstone in understanding complex functions and their tendencies.