Problem 37 - Calculus: Early Transcendentals

Question

Limit proofs Use the precise definition of a limit to prove the following limits. Specify a relationship between \(\varepsilon\) and \(\delta\) that guarantees the limit exists. \(\lim _{x \rightarrow 0}\left(x^{2}+x^{4}\right)=0\) (Hint: You may use the fact that if \(|x| < c\) then \(\left.x^{2}< c^{2} .\right)\)

Step-by-Step Solution

Answer

Question: Prove that \(\lim _{x \rightarrow 0}\left(x^{2}+x^{4}\right)=0\) using the precise definition of a limit and the relationship between \(\varepsilon\) and \(\delta\).

Step 1: Apply the definition of limit

According to the precise definition of a limit, we need to show that for any \(\varepsilon > 0\), there exists a \(\delta > 0\) such that \(0<|x-0|<\delta\) implies \(|x^2 + x^4 - 0|<\varepsilon\).

Step 2: Bounding the terms

Since we want to find \(|x^2 + x^4 - 0|\), we can use the hint to bound the terms. We can write: \(|x^2 + x^4| \leq |x^2| + |x^4|\). Now, if we assume \(|x|

Step 3: Choose c as \(\delta\) and find the relationship between \(\varepsilon\) and \(\delta\)

We will choose \(c\) as \(\delta\). Therefore, if \(0<|x-0|<\delta\), we have: \(|x^2 + x^4| \leq \delta^2 + \delta^4\). Now, we need to make sure that this expression is less than \(\varepsilon\). Thus, we set: \(\delta^2 + \delta^4 < \varepsilon\). Our goal is to find a relationship between \(\varepsilon\) and \(\delta\) that guarantees the limit exists.

Step 4: Apply the condition that \(\delta < 1\)

If we assume \(\delta < 1\), this simplifies our expression: \(\delta^2 + \delta^4 < \delta^2 + \delta^2 = 2\delta^2\). Now, if we choose \(\delta = \min\{1, \sqrt{\frac{\varepsilon}{2}}\}\), we will have: \(2\delta^2 \leq 2\left(\frac{\varepsilon}{2}\right) = \varepsilon\). Thus, we have found the relationship between \(\varepsilon\) and \(\delta\): \(\delta = \min\{1, \sqrt{\frac{\varepsilon}{2}}\}\).

Step 5: Conclusion

By using the precise definition of a limit and finding the relationship between \(\varepsilon\) and \(\delta\) as \(\delta = \min\{1, \sqrt{\frac{\varepsilon}{2}}\}\), we have proven that: $$\lim _{x \rightarrow 0}\left(x^{2}+x^{4}\right)=0.$$

Key Concepts

Precise Definition of a Limit

The precise definition of a limit helps us rigorously verify that a function approaches a certain value as the input approaches some point. It demands that for every positive number, \(\varepsilon\), no matter how small, there is a corresponding positive number, \(\delta\), such that as the input \(x\) gets within a distance \(\delta\) from \(a\), the function value \(f(x)\) gets within \(\varepsilon\) from \(L\), the limit value.

Formally, we express it as: for all \(\varepsilon > 0\), there exists a \(\delta > 0\) such that whenever \(0 < |x - a| < \delta\), it follows that \(|f(x) - L| < \varepsilon\).
In the context of \( \lim_{x \to 0} (x^2 + x^4) = 0 \), this means we need to show how close \(x^2 + x^4\) can get to 0 as \(x\) approaches 0, following the epsilon-delta criteria.

Relationship between \(\varepsilon\) and \(\delta\)

Finding a connection between \(\varepsilon\) and \(\delta\) is a crucial step for proving limits. This relationship acts as a bridge that links how close the function value needs to be to the limit (\(\varepsilon\)) with how close the input \(x\) needs to be to the point of interest (\(\delta\)).

In our example, we ended up needing \(x\) to be so close to 0 such that the squared and fourth power terms amounted to less than \(\varepsilon\). Through the steps, after exploring bounds and substituting wisely, we found that choosing \( \delta = \min \{1, \sqrt{\varepsilon/2} \} \) ensures the function stays within the desired \(\varepsilon\) limit range.
This approach makes it critical to think strategically about what values \(\delta\) should take on, allowing the proper handling of complex expressions.

Bounding Terms in Limits

Bounding is an effective strategy to tackle expressions in limit proofs, particularly when dealing with polynomial terms like \(x^2 + x^4\). Here, bounding establishes a way to simplify and control the behavior of the function as \(x\) nears a specific point.

Given \(|x| < c\), bounding is achieved by:\

\
• First separating each term: \(|x^2 + x^4| \leq |x^2| + |x^4|\)\
\
• Then applying known inequalities: \(|x^2| < c^2\) and \(|x^4| < c^4\)\
\
• Consolidating them to find \(|x^2 + x^4| < c^2 + c^4\)\
\

Mounting these bounds assists in setting up an inequality that will ease linking to the \(\varepsilon\), creating a pivotal step to manage how close the function must be to zero.

Epsilon-Delta Proof

An epsilon-delta proof uses the rigorous \(\varepsilon\)-\(\delta\) definition to establish that a function approaches a given limit when \(x\) approaches a certain point. It’s the heart of demonstrating that a certain sequence or function truly converges.

Successfully constructing such proof involves:

\
• Establishing bounds to control and simplify the behavior of the function’s value\
\
• Discovering a precise relation between \(\varepsilon\) and \(\delta\), which dictates their dependencies\
\
• Verifying the condition holds true as per the definition for each \(\varepsilon\) by selecting the appropriate \(\delta\).\
\

In our worked example, defining \(\delta = \min \{1, \sqrt{\varepsilon/2} \} \) sufficed to meet the conditions, neatly bounding the terms in accordance with \(\varepsilon\) to ensure the limit of \(x^2 + x^4\) approached zero. This structured methodology guarantees the proof holds under all allowable values.

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Key Concepts

Precise Definition of a Limit

The precise definition of a limit helps us rigorously verify that a function approaches a certain value as the input approaches some point. It demands that for every positive number, \(\varepsilon\), no matter how small, there is a corresponding positive number, \(\delta\), such that as the input \(x\) gets within a distance \(\delta\) from \(a\), the function value \(f(x)\) gets within \(\varepsilon\) from \(L\), the limit value.

Formally, we express it as: for all \(\varepsilon > 0\), there exists a \(\delta > 0\) such that whenever \(0 < |x - a| < \delta\), it follows that \(|f(x) - L| < \varepsilon\).
In the context of \( \lim_{x \to 0} (x^2 + x^4) = 0 \), this means we need to show how close \(x^2 + x^4\) can get to 0 as \(x\) approaches 0, following the epsilon-delta criteria.

Relationship between \(\varepsilon\) and \(\delta\)

Finding a connection between \(\varepsilon\) and \(\delta\) is a crucial step for proving limits. This relationship acts as a bridge that links how close the function value needs to be to the limit (\(\varepsilon\)) with how close the input \(x\) needs to be to the point of interest (\(\delta\)).

In our example, we ended up needing \(x\) to be so close to 0 such that the squared and fourth power terms amounted to less than \(\varepsilon\). Through the steps, after exploring bounds and substituting wisely, we found that choosing \( \delta = \min \{1, \sqrt{\varepsilon/2} \} \) ensures the function stays within the desired \(\varepsilon\) limit range.
This approach makes it critical to think strategically about what values \(\delta\) should take on, allowing the proper handling of complex expressions.

Bounding Terms in Limits

Bounding is an effective strategy to tackle expressions in limit proofs, particularly when dealing with polynomial terms like \(x^2 + x^4\). Here, bounding establishes a way to simplify and control the behavior of the function as \(x\) nears a specific point.

Given \(|x| < c\), bounding is achieved by:\

\
• First separating each term: \(|x^2 + x^4| \leq |x^2| + |x^4|\)\
\
• Then applying known inequalities: \(|x^2| < c^2\) and \(|x^4| < c^4\)\
\
• Consolidating them to find \(|x^2 + x^4| < c^2 + c^4\)\
\

Mounting these bounds assists in setting up an inequality that will ease linking to the \(\varepsilon\), creating a pivotal step to manage how close the function must be to zero.

Epsilon-Delta Proof

An epsilon-delta proof uses the rigorous \(\varepsilon\)-\(\delta\) definition to establish that a function approaches a given limit when \(x\) approaches a certain point. It’s the heart of demonstrating that a certain sequence or function truly converges.

Successfully constructing such proof involves:

\
• Establishing bounds to control and simplify the behavior of the function’s value\
\
• Discovering a precise relation between \(\varepsilon\) and \(\delta\), which dictates their dependencies\
\
• Verifying the condition holds true as per the definition for each \(\varepsilon\) by selecting the appropriate \(\delta\).\
\

In our worked example, defining \(\delta = \min \{1, \sqrt{\varepsilon/2} \} \) sufficed to meet the conditions, neatly bounding the terms in accordance with \(\varepsilon\) to ensure the limit of \(x^2 + x^4\) approached zero. This structured methodology guarantees the proof holds under all allowable values.