Problem 30 - Calculus Early Transcendentals

Question

Use the precise definition of infinite limits to prove the following limits. $$\lim _{x \rightarrow-1} \frac{1}{(x+1)^{4}}=\infty$$

Step-by-Step Solution

Answer

Question: Use the precise definition of infinite limits to prove that: $$\lim _{x \rightarrow-1} \frac{1}{(x+1)^{4}}=\infty$$ Answer: We showed that for any given M > 0, there exists a δ > 0 such that if 0 < |x + 1| < δ, then f(x) > M, where f(x) = 1/(x + 1)^4. We determined the value of δ as: $$\delta=\sqrt[4]{\frac{1}{M}}$$ With this, we were able to prove the given limit using the precise definition of infinite limits.

Step 1: Set up the problem

First, let's rewrite the given limit as follows: $$\lim _{x \rightarrow-1} \frac{1}{(x+1)^{4}}=\infty$$ We want to prove that for any given M > 0, there exists a δ > 0 such that if 0 < |x + 1| < δ, then f(x) > M, where f(x) = 1/(x + 1)^4.

Step 2: Choose an arbitrary M > 0

Let M be any positive real number (M > 0).

Step 3: Set the inequality f(x) > M

Now, we'll set up the inequality involving f(x) and M: $$\frac{1}{(x+1)^{4}}>M$$

Step 4: Solve for x

In order to find the appropriate δ, we need to solve this inequality for x. First, let's take the reciprocal of both sides, keeping in mind that taking the reciprocal reverses the inequality sign: $$(x+1)^{4}<\frac{1}{M}$$ Now, by taking the 4th root, we obtain: $$|x+1|<\sqrt[4]{\frac{1}{M}}$$

Step 5: Determine the value of δ

From the previous step, we see that the value of δ should be: $$\delta=\sqrt[4]{\frac{1}{M}}$$

Step 6: Proof

Now we can show that our chosen δ works for the given limit. Given any M > 0, we have found a δ > 0 such that: $$\delta=\sqrt[4]{\frac{1}{M}}$$ If 0 < |x + 1| < δ, then $$\frac{1}{(x+1)^{4}}>M$$ So, the limit can be written as follows: $$\lim _{x \rightarrow-1} \frac{1}{(x+1)^{4}}=\infty$$ Thus, we have successfully proven the given limit using the precise definition of infinite limits.

Key Concepts

Precise Definition of Limits

The concept of limits in calculus is fundamental for understanding how functions behave as they approach certain points. Infinite limits specifically describe scenarios where a function grows without bound as the input approaches a particular value. In simpler terms, the function's output heads towards positive or negative infinity. This needs a precise mathematical definition to be proven correctly.

For a function to have an infinite limit at a point, say \( x = c \), we need to show that, for any arbitrarily large positive number \( M \), there exists a distance \( \delta \) that ensures if the input \( x \) is within \( \delta \) of \( c \) (but not equal to \( c \)), the function's output \( f(x) \) will exceed \( M \).

In the exercise provided, we aimed to prove that \( \lim _{x \rightarrow-1} \frac{1}{(x+1)^{4}}=\infty \). This involves showing that given any large number \( M > 0 \), we can find a \( \delta > 0 \) ensuring that whenever \( x \) is close to \(-1\) (closer than \( \delta \)), \( f(x) \) becomes greater than \( M \). This process illustrates the rigorous approach needed to confirm that functions indeed tend to infinity.

Limit Proofs

Proving limits, especially infinite limits, requires substantiation through precise and logical steps. The exercise shows the use of such a structured approach to prove the statement \( \lim _{x \rightarrow-1} \frac{1}{(x+1)^{4}}=\infty \).

Let's break down the proof process:

• **Set up the inequality:** Transforming the limit statement into an inequality helps in visualizing how the function behaves. Here, starting with \( \frac{1}{(x+1)^{4}}>M \) simplifies understanding.


• **Solve the inequality:** Next is to manipulate this inequality logically to solve for \( x \). The key here was to use reciprocal and root properties which shifted the inequality to \( |x+1|<\sqrt[4]{\frac{1}{M}} \).


• **Find \( \delta \):** This results in a \( \delta \) value ensuring that the function indeed grows beyond any \( M \) when \( x \) is within this bound. In our context, \( \delta = \sqrt[4]{\frac{1}{M}} \)

These steps craft a logical argument to authenticate the limit statement. It converts the abstract concept of limits into a tangible pathway to comprehend and demonstrate using mathematical thought processes.

Inequalities in Calculus

Inequalities are crucial in calculus, especially when dealing with limits and proving the behavior of functions. Calculus frequently employs inequalities as tools to encapsulate behaviors like growth at infinite, bounding functions, and confirming continuity or limits.

In the exercise, we used inequalities to manage the function \( \frac{1}{(x+1)^{4}} \). Knowing how to handle inequalities is crucial:

• **Inequality Manipulation:** It's necessary to be comfortable with flipping inequality signs especially when dealing with reciprocals or even roots. For instance, turning \( \frac{1}{(x+1)^{4}}>M \) into \( (x+1)^{4}<\frac{1}{M} \).


• **Algebraic Manipulations:** These are used to isolate the variable through steps like reciprocation or taking roots, as shown when simplifying \( |x+1|<\sqrt[4]{\frac{1}{M}} \).

Thus, adeptly handling inequalities allows for precise definitions and confirmation of limits, which makes them an indispensable aspect of calculus, underpinning a wide range of proofs and theorems.