Problem 26

Question

$$ \text { Prove that } \lim _{x \rightarrow 0^{+}} \sqrt{x}=0 \text { . } $$

Step-by-Step Solution

Verified

Answer

The limit is 0 because $ \sqrt{x} < \varepsilon $ for $ x < \varepsilon^2 $.

1Step 1: Understanding the Limit Definition

To prove that $ \lim_{{x \to 0^+}} \sqrt{x} = 0 $, we need to use the $ \varepsilon-\delta $ definition of a limit. This definition states that for every $ \varepsilon > 0 $, there exists a $ \delta > 0 $ such that if $ 0 < x < \delta $, then $ \left| \sqrt{x} - 0 \right| < \varepsilon $.

2Step 2: Express the Function in Inequality Form

We need to ensure that $ \left| \sqrt{x} - 0 \right| = \sqrt{x} < \varepsilon $. This simplifies to finding a $ \delta $ for which $ \sqrt{x} < \varepsilon $ when $ 0 < x < \delta $.

3Step 3: Finding Delta in Terms of Epsilon

Since $ \sqrt{x} < \varepsilon $ implies $ x < \varepsilon^2 $, we can choose $ \delta = \varepsilon^2 $. This means that if $ 0 < x < \varepsilon^2 $, then $ \sqrt{x} < \varepsilon $.

4Step 4: Conclusion of the Proof

We have shown that for any $ \varepsilon > 0 $, choosing $ \delta = \varepsilon^2 $ guarantees that if $ 0 < x < \delta $, then $ \sqrt{x} < \varepsilon $. Therefore, by the definition of a limit, $ \lim_{{x \to 0^+}} \sqrt{x} = 0 $.

Key Concepts

LimitsEpsilon-Delta DefinitionContinuitySquare Root Function

Limits

Limits are a fundamental concept in calculus, allowing us to understand the behavior of functions as they approach specific points or infinity. When we say $ \lim_{{x \to a}} f(x) = L $, it means that as $ x $ approaches $ a $, the function $ f(x) $ gets arbitrarily close to the limit $ L $. This concept is crucial to analyze functions at points where they might not be directly defined.

Here are some important aspects of limits:

Approaching a Point: Limits focus on values approaching a particular point, not necessarily the value at that point.
Direction Matters: Limits can be one-sided, approaching from the right ($ x \rightarrow a^{+} $) or from the left ($ x \rightarrow a^{-} $).
Behavior Understanding: They help us understand function behavior, including at points of discontinuity.

Understanding limits lays a solid foundation for exploring deeper concepts like derivatives and integrals in calculus.

Epsilon-Delta Definition

The epsilon-delta definition of a limit provides a rigorous way to understand what we mean by a function approaching a particular value. This definition is represented as:For every $ \varepsilon > 0 $, there exists a $ \delta > 0 $ such that if $ 0 < |x - a| < \delta $, then $ |f(x) - L| < \varepsilon $.

This means:

Epsilon ($ \varepsilon $): Represents how close $ f(x) $ needs to be to the limit $ L $.
Delta ($ \delta $): Indicates how close $ x $ must be to $ a $ to ensure $ f(x) $ is within $ \varepsilon $ of $ L $.
Precision Control: For any precision $ \varepsilon $, you can find a corresponding $ \delta $ showing the function's limits.

This definition is key for proving limit statements mathematically, as seen in the solution where $ \delta $ was chosen based on $ \varepsilon $ for the function $ \sqrt{x} $.

Continuity

Continuity of a function at a point implies that as you zoom in at the point, there is no sudden jump, gap, or break. Mathematically, a function $ f(x) $ is continuous at a point $ a $ if:

$ f(a) $ is defined,
$ \lim_{{x \to a}} f(x) $ exists,
$ \lim_{{x \to a}} f(x) = f(a) $.

These conditions ensure that the behavior of the function smoothly aligns with the value at $ a $.

Connections:

Limits: If a limit does not exist, a function cannot be continuous at that point.
Epsilon-Delta Definition: This same approach can establish the continuity by proving the limit equals the function's value at the point.

Working with continuity, particularly with limits and specific functions like the square root, gives deeper insights into function behavior across domains.

Square Root Function

The square root function, represented as $ f(x) = \sqrt{x} $, is one of the basic elementary functions. It maps each non-negative real number $ x $ to its square root, $ x^{1/2} $. This function is notable for its unique characteristics and behavior. Here are some features:

Domain: The function is only defined for $ x \ge 0 $ since square roots of negative numbers are not real.
Increasing Nature: As $ x $ increases, $ \sqrt{x} $ also increases, which is evident from its positive derivative.
Continuity: $ \sqrt{x} $ is continuous where it is defined, providing a smooth curve in the graph.

Square roots also relate closely to the discussed topics:

Limits: Understanding $ \lim_{{x \to 0^+}} \sqrt{x} = 0 $ involves analyzing the function as it approaches zero.
Epsilon-Delta: Establishes how the values of $ \sqrt{x} $ approach zero from the right using precise methods.

By navigating through these concepts, students can gain a robust understanding of both the square root function and its role in calculus.

Problem 26

Other exercises in this chapter

Problem 26

Find the limits. $$ \lim _{n \rightarrow \infty} \frac{n^{2}}{\sqrt{n^{3}+2 n+1}} $$

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Problem 26

, find the limits if $\lim _{x \rightarrow a} f(x)=3$ and $\lim _{x \rightarrow a} g(x)=-1($ see Example 4) $.$ $$ \lim _{x \rightarrow a} \frac{2 f(x)-3

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Problem 26

, use a calculator to find the indicated limit. Use a graphing calculator to plot the function near the limit point. $$ \lim _{t \rightarrow 0} \frac{1-\cot t}{

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Problem 27

What points, if any, are the functions discontinuous? $$ r(\theta)=\tan \theta $$

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