Problem 55 - Calculus: Early Transcendentals

Question

Precise definitions for left- and right-sided limits Use the following definitions. Assume \(f\) exists for all \(x\) near a with \(x>a\). We say that the limit of \(f(x)\) as \(x\) approaches a from the right of a is \(L\) and write \(\lim _{x \rightarrow a^{+}} f(x)=L,\) iffor any \(\varepsilon> 0 \) there exists \(\delta > 0\) such that $$|f(x)-L| < \varepsilon \text { whenever } 0 < x- a < \delta $$ Assume \(f\) exists for all \(x\) near a with \(x < a .\) We say that the limit of \(f(x)\) as \(x\) approaches a from the left of a is \(L\) and write \(\lim _{x \rightarrow a^{-}} f(x)=L,\) iffor any \(\varepsilon > 0 \) there exists \(\delta > 0 \) such that $$ |f(x)-L| < \varepsilon \text { whenever } 0 < a - x < \delta $$ One-sided limit proof Prove that \(\lim _{x \rightarrow 0^{+}} \sqrt{x}=0\)

Step-by-Step Solution

Answer

Question: Prove that the one-sided limit \(\lim_{x \rightarrow 0^{+}} \sqrt{x}=0\). Answer: To prove that \(\lim_{x \rightarrow 0^{+}} \sqrt{x}=0\), we showed that for any \(\varepsilon > 0\), there exists a \(\delta > 0\) such that \(|\sqrt{x}-0| < \varepsilon\) for all \(x\) in the interval \((0, \delta)\). By choosing \(\delta = \varepsilon^2\), we demonstrated that this condition holds for any arbitrarily small \(\varepsilon\), and therefore, we can conclude that \(\lim _{x \rightarrow 0^{+}} \sqrt{x} = 0\).

Step 1: Set up the problem

We want to prove that \(\lim _{x \rightarrow 0^{+}} \sqrt{x}=0\). According to the definition of the right-sided limit, for any \(\varepsilon > 0\), there exists \(\delta > 0\) such that $$|\sqrt{x} - 0| < \varepsilon \text{ whenever } 0 < x - 0 < \delta.$$ We can rewrite this inequality as: $$|\sqrt{x}| < \varepsilon \text{ whenever } 0 < x < \delta.$$

Step 2: Find an appropriate \(\delta\)

To find a suitable \(\delta\), we can square both sides of the inequality \(|\sqrt{x}| < \varepsilon\): $$|\sqrt{x}|^2 < \varepsilon^2$$ $$|x| < \varepsilon^2$$ Since \(x>0\), we have: $$x < \varepsilon^2$$ This suggests that we can choose \(\delta = \varepsilon^2\).

Step 3: Prove the statement using the chosen \(\delta\)

Now let's verify that our choice of \(\delta\) satisfies the definition of the right-sided limit. We want to show that for any \(\varepsilon > 0\), if \(0 < x - 0 < \delta\), then \(|\sqrt{x}| < \varepsilon\). Suppose \(0 < x - 0 < \delta\), where \(\delta = \varepsilon^2\). Then we have \(0 < x < \varepsilon^2\). Since \(x>0\), taking the square root of both sides doesn't change the direction of the inequality: $$0 < \sqrt{x} < \varepsilon$$ Therefore, \(|\sqrt{x} - 0| = |\sqrt{x}| < \varepsilon\) whenever \(0 < x < \delta\).

Step 4: Conclusion

We have shown that for any arbitrarily small \(\varepsilon > 0\), there exists a \(\delta = \varepsilon^2\) such that \(|\sqrt{x} - 0| < \varepsilon\) whenever \(0 < x < \delta\). Thus, by definition, we can conclude that \(\lim _{x \rightarrow 0^{+}} \sqrt{x} = 0\).

Key Concepts

Right-Sided Limit

Understanding the concept of a right-sided limit is essential in calculus when we are interested in how a function behaves as it approaches a particular point from the right side. Suppose you have a function \( f(x) \) and you are examining its behavior as \( x \) nears a specific point \( a \) from the right (i.e., \( x > a \)). The notation used for this is \( \lim_{x \to a^+} f(x) = L \), indicating that as \( x \) makes its way toward \( a \) from the right, the function \( f(x) \) is getting closer and closer to \( L \).

One crucial element to establish this limit is the \( \varepsilon-\delta \) definition. When working with a right-sided limit, we can say that the limit holds true if for any small positive \( \varepsilon \), there exists some positive \( \delta \) such that whenever \( 0 < x - a < \delta \), the condition \( |f(x) - L| < \varepsilon \) is satisfied. This means that you can make the values of \( f(x) \) as close to \( L \) as you want by choosing \( x \) sufficiently close to \( a \) from the right.

• The emphasis of a right-sided limit is to ensure that all considerations are for \( x > a \).
• This type of limit is particularly useful in scenarios where a function may behave differently at \( a \) than on one side or the other.

Left-Sided Limit

The left-sided limit is almost identical in concept to the right-sided limit, but instead, it considers the approach of \( x \) from the left side of \( a \). This is crucial in situations where the behavior of \( f(x) \) might change depending on which direction you are approaching \( a \).

The notation for a left-sided limit is \( \lim_{x \to a^-} f(x) = L \). This indicates that as \( x \) approaches \( a \) from values less than \( a \) (so \( x < a \)), the function \( f(x) \) is getting closer to \( L \).

Using the \( \varepsilon-\delta \) definition, the limit \( \lim_{x \to a^-} f(x) = L \) holds if, for every small positive \( \varepsilon \), there is a positive \( \delta \) such that whenever \( 0 < a - x < \delta \), the inequality \( |f(x) - L| < \varepsilon \) is satisfied.

• Here, \( x < a \) ensures we're strictly considering the left side.
• Understanding both right-sided and left-sided limits is crucial for ensuring the overall limit exists and behaves consistently in all approaches.

Epsilon-Delta Definition

The \( \varepsilon-\delta \) definition is the backbone of the formal definition of limits, providing a precise way to state what it means for a function to approach a limit as the input approaches a point. It's vital for proving statements about both one-sided and two-sided limits.

In the context of one-sided limits, like right-sided and left-sided limits, the \( \varepsilon-\delta \) definition holds by stating that for any \( \varepsilon > 0 \), however small, there must be a suitable \( \delta > 0 \) such that if the input value of \( x \) satisfies \( 0 < |x - a| < \delta \), then the function's value \( f(x) \) will satisfy \( |f(x) - L| < \varepsilon \).

This definition helps in establishing rigorous proofs for limit statements.

• Finding an appropriate \( \delta \) for given \( \varepsilon \) often involves manipulating the function \( f(x) \) to fit within the \( \varepsilon \) boundary.
• Once \( \delta \) is found, one can confidently state and prove limit claims based on this exact framework, ensuring precision and accuracy in calculus.