Question

In Exerises 2.1.28-2.1.30 consider only the case where at least one of \(L_{1}\) and \(L_{2}\) is \(\pm \infty\). (a) Prove: If \(\lim _{x \rightarrow x_{0}} f(x)=L_{1} \cdot \lim _{x \rightarrow x_{0}} g(x)=L_{2} \neq 0,\) and \(L_{1} / L_{2}\) is not indeterminate, then $$ \lim _{x \rightarrow x_{0}}\left(\frac{f}{g}\right)(x)=\frac{L_{1}}{L_{2}} $$ (b) Show that it is necessary to assume that \(L_{2} \neq 0\) in \((a)\) by considering \(f(x)=\) \(\sin x, g(x)=\cos x,\) and \(x_{0}=\pi / 2\)

Short answer

Question: Prove that if \(\lim_{x \rightarrow x_{0}} f(x) = L_{1}\) and \(\lim_{x \rightarrow x_{0}} g(x) = L_{2} \neq 0\), then \(\lim_{x \rightarrow x_{0}} (\frac{f}{g})(x) = \frac{L_{1}}{L_{2}}\). Also, discuss the necessity of the condition L2 ≠ 0.

Step-by-step solution

Understand the given conditions and problem

We are given two functions, f(x) and g(x), with known limits L1 and L2 at x approaches x0. The limits L1 and L2 can be positive infinity, negative infinity or any real number except zero. We need to prove that under the given conditions, the limit of the function f(x)/g(x) as x approaches x0 is equal to L1/L2.

Prove the limit of f(x)/g(x) under given condition:

To prove the limit of f(x)/g(x), we will use the limit laws for fractions as follows: i) If \lim_{x \to x_{0}} f(x) = L_{1} \neq 0 and \lim_{x \to x_{0}} g(x) = L_{2} \neq 0, then \lim_{x \to x_{0}} \frac{f(x)}{g(x)} = \frac{\lim_{x \to x_{0}} f(x)}{\lim_{x \to x_{0}} g(x)} = \frac{L_{1}}{L_{2}}. Now, under the given conditions, we can apply limit laws for fractions and directly get the result \lim_{x \rightarrow x_{0}} (\frac{f}{g})(x) = \frac{L_{1}}{L_{2}}.

Show the necessity of L2 ≠ 0 by taking a specific case

In this part, we have to show the importance of the given condition L2 ≠ 0. We will take f(x) = sin(x) and g(x) = cos(x) with x0 = π/2. Now, we have: 1) \(\lim_{x \rightarrow \frac{\pi}{2}} f(x) = \lim_{x \rightarrow \frac{\pi}{2}} \sin(x) = 1 = L_{1}\) 2) \(\lim_{x \rightarrow \frac{\pi}{2}} g(x) =\lim_{x \rightarrow \frac{\pi}{2}} \cos(x) = 0 = L_{2}\) From the above results, we can see that L2 = 0. Now, let's compute the limit of \(\frac{f(x)}{g(x)}\) as x approaches x0: \(\lim_{x \rightarrow \frac{\pi}{2}} \frac{f(x)}{g(x)} = \lim_{x \rightarrow \frac{\pi}{2}} \frac{\sin(x)}{\cos(x)}\) As we can see, the above limit is not defined as g(x) approaches zero (dividing by zero is undefined). Thus, it is necessary to assume that L2 ≠ 0 in (a).

Key concepts

Real Analysis

Real analysis is a branch of mathematics that deals with the properties of real numbers and the analytical properties of real functions and sequences. It plays a crucial role in understanding the behavior of functions and their limits. When dealing with limits, it is essential to characterize not just how functions behave near a point, but also to formalize our intuition regarding convergence, continuity, and divergence within a rigorous mathematical framework.

For example, it specifies the conditions under which we can say that a function approaches a certain value as the input approaches a certain point. This has vast implications in calculus and mathematical analysis since it forms the foundation of derivative and integral concepts. Understanding limits through real analysis allows students to navigate challenging problems and enables them to rigorously justify their reasoning when evaluating the behavior of functions.

Limit Laws

Limit laws are a set of rules that allow us to compute the limit of a function using the limits of its simpler components. For instance, the limit of a sum, product, or quotient of functions can often be determined by taking the limit of each part separately. An important aspect is that these laws only apply if these component limits exist and are finite.

For example, if we know the limits as `x` approaches `x_0` for two functions `f(x)` and `g(x)`, and these limits are `L_1` and `L_2` respectively, limit laws let us deduce that the limit of `f(x) + g(x)` as `x` approaches `x_0` is `L_1 + L_2`, assuming `L_1` and `L_2` are not infinite. Similarly, the limit law for division used in the exercise allows us to divide the limits of two functions, which aids in proving that the limit of a ratio is the ratio of the limits, provided the denominator does not approach zero.

Indeterminate Forms

When evaluating limits, we sometimes encounter indeterminate forms such as `0/0` or `∞/∞`. These expressions do not have a clear, definite value and require further analysis to determine the limit.

Indeterminate forms are significant because they often arise in calculus, particularly in the evaluation of derivatives using L'Hôpital's Rule or in the computation of limits of functions involving algebraic manipulation. They show up in situations where the behaviors of the numerator and denominator of a fraction conflict with each other as they approach the limit, which makes the outcome uncertain without further investigation. Recognizing indeterminate forms helps students to apply appropriate techniques to resolve them and find the true limit of a function.

Infinite Limits

Infinite limits describe the behavior of functions as they increase or decrease without bound as the input approaches a certain value. Unlike finite limits, where functions approach a specific number, an infinite limit means the function grows towards positive or negative infinity.

These limits are important when assessing the unbounded behavior of functions, and they come with their own unique set of rules. For example, any time we divide a non-zero number by an infinitesimally small positive or negative number, the result is an infinite limit. However, when dealing with infinite limits, caution must be exercised; the presence of an infinite limit such as `L_1` being `∞` or `-∞` changes how we apply limit laws, leading to different forms of proof and logic compared to finite limits.