Question

Prove: If \(\lim _{x \rightarrow x_{0}} f(x)=L>0\), then \(\lim _{x \rightarrow x_{0}} \sqrt{f(x)}=\sqrt{L}\).

Short answer

Question: Prove that if the limit of a function f(x) as x approaches x₀ is L, where L > 0, then the limit of the square root of the function, sqrt(f(x)), as x approaches x₀ is equal to the square root of L. Answer: To prove this, we defined a new function g(x) = sqrt(f(x)) and transformed the expression |g(x) - sqrt(L)| into a form that can be worked with the definition of the limit of f(x), which is |f(x) - L| < ε₁. Choosing ε₁ = ε|sqrt(f(x)) + sqrt(L)| and substituting it back into the expression for the limit, we have shown that |g(x) - sqrt(L)| < ε for all ε > 0. Thus, we have proven that the limit of sqrt(f(x)) as x approaches x₀ is equal to the square root of L.

Step-by-step solution

Understand the definition of limit

Given the limit of a function, we can express the definition of the limit as follows: $$\lim_{x \rightarrow x_0} f(x) = L$$ if, for every \(\epsilon > 0\), there exists \(\delta > 0\) such that if \(0 < |x - x_0| < \delta\), then \(|f(x) - L| < \epsilon\). Now, we want to prove that \(\lim_{x \rightarrow x_0} \sqrt{f(x)} = \sqrt{L}\).

Define a new function

First, let us define a new function \(g(x) = \sqrt{f(x)}\). Then, our task is to prove that: $$\lim_{x \rightarrow x_0} g(x) = \sqrt{L}$$

Express the limit of the new function

We want to express the limit of the new function in terms of ε and δ. Following the definition of the limit, we need to show that for every \(\epsilon > 0\), there exists \(\delta > 0\) such that if \(0 < |x - x_0| < \delta\), then \(|g(x) - \sqrt{L}| < \epsilon\).

Utilize properties of limit and square root

We have \(|g(x) - \sqrt{L}| = |\sqrt{f(x)} - \sqrt{L}|\). By using the properties of the limit and square root, we can write this as: $$|\sqrt{f(x)} - \sqrt{L}| = \frac{|f(x) - L|}{|\sqrt{f(x)} + \sqrt{L}|}$$ Now, we have the equation in a form that we can work with the definition of the limit of \(f(x)\). Since we know \(\lim_{x \rightarrow x_0} f(x) = L\), we have: If \(0 < |x - x_0| < \delta\), then \(|f(x) - L| < \epsilon_1\), where \(\epsilon_1 > 0\). We can now choose \(\epsilon_1\) such that: $$\epsilon_1 = \epsilon |\sqrt{f(x)} + \sqrt{L}|$$

Substitute ε₁ to prove the limit

Now substitute the expression for \(\epsilon_1\) into the definition of limit for \(f(x)\): If \(0 < |x - x_0| < \delta\), then \(|f(x) - L| < \epsilon |\sqrt{f(x)} + \sqrt{L}|\). Now, we can divide both sides by \(|\sqrt{f(x)} + \sqrt{L}|\) to get: $$\frac{|f(x) - L|}{|\sqrt{f(x)} + \sqrt{L}|} < \epsilon$$ We have now shown that if \(0 < |x-x_0| < \delta\), then \(|g(x) - \sqrt{L}| = \frac{|f(x) - L|}{|\sqrt{f(x)} + \sqrt{L}|} < \epsilon\). Therefore, we have proven that: $$\lim_{x \rightarrow x_0} \sqrt{f(x)} = \lim_{x \rightarrow x_0} g(x) = \sqrt{L}$$.

Key concepts

Limit of a Function

The concept of a limit is an essential building block in calculus, particularly when trying to understand continuous behavior of functions as they approach a specific input value. In simple terms, saying the limit of a function, let's call it f(x), equals some value L as x approaches x₀, is mathematically notated as:
\[\lim_{{x \rightarrow x_0}} f(x) = L\]
What this means in a practical sense is that as we choose values of x closer and closer to x₀, the function's output gets closer and closer to L. However, it's important to note that this doesn't mean f(x) will actually reach L when x is exactly x₀. This concept is foundational when proving more complex limit properties, such as those involving square roots.

Epsilon-Delta Definition of Limit

The epsilon-delta definition forms the precise 'language' of limits. To say that the limit of f(x) as x approaches x₀ is L, you show that for every positive number epsilon (ε), no matter how small, there's a corresponding positive number delta (δ) where if x is within δ of x₀ (excluding x₀ itself), then f(x) will be within ε of L. If you can find such a δ for every ε given, then L is indeed the limit of f(x) as x approaches x₀. This criterion is used to rigorously confirm the behavior of functions near a point and forms a logical basis for proving limits involving square roots.

Properties of Limits

Limits obey several properties that allow us to manipulate and analyze them effectively. For instance, if limits of f(x) and g(x) exist as x approaches x₀ and are equal to L and M respectively, then the limit of f(x) + g(x) as x approaches x₀ equals L + M. Moreover, the limit of a constant times a function is the constant times the limit of the function. These properties are particularly useful when working with compound functions, such as those including square roots, since they allow us to break down complex limit problems into simpler parts that can be tackled individually.

Square Roots and Limits

When dealing with limits involving square roots, the properties of square roots become crucial. Notably, for any real numbers a and b that are non-negative, the square root of their product is the product of their square roots, while the square root of a quotient is the quotient of their square roots. This helps when handling the limit of a square root function by enabling us to express differences involving square roots in a way that we can apply the epsilon-delta definition, just like our original function f(x). It can be particularly tricky because the presence of square roots may introduce absolute values that must be dealt with carefully, as shown in the step-by-step solution where the original problem was manipulated to find a suitable epsilon-delta proof for the limit involving a square root function.