Problem 98 - Single Variable Calculus

Question

Show that \(\lim _{x \rightarrow a} \cos x=\cos a\). (See the hint for Exercise 97.)

Step-by-Step Solution

Answer

To show that \(\lim_{x \rightarrow a} \cos x = \cos a\), we used the limit definition and introduced the auxiliary function g(x) = cos(x). We rewrote \(|cos(x) - cos(a)|\) using a trigonometric identity involving the sine function: \[|cos(x) - cos(a)| = |-2 sin(\frac{x - a}{2}) sin(\frac{x + a}{2})|\]. Applying inequalities for the sine function and using the facts that the sine function is continuous and differentiable, we showed that for every ε > 0, there exists δ > 0, such that if 0 < |x - a| < δ, then \(-2 ε \leq -2 sin(\frac{x - a}{2}) sin(\frac{x + a}{2}) \leq 2 ε\), thus proving that \(\lim_{x \rightarrow a} \cos x = \cos a\).

Step 1: Recall Limit Definition

Recall the limit definition: If \[ \lim _{x \rightarrow a} f(x) = L\], then for every ε > 0, there exists a δ > 0, such that if 0 < |x - a| < δ, then |f(x) - L| < ε. In our case, we have to show that for every ε > 0, there exists a δ > 0, such that if 0 < |x - a| < δ, then |cos(x) - cos(a)| < ε.

Step 2: Introduce Auxiliary Function

To proof the limit, let's use the auxiliary function g(x) = cos(x), then \[|g(x) - g(a)| = |cos(x) - cos(a)|\]

Step 3: Use Trigonometric Identity for |cos(x) - cos(a)|

We can rewrite |cos(x) - cos(a)| using a trigonometric identity that involves the sine function, which is: \[|cos(x) - cos(a)| = |-2 sin(\frac{x - a}{2}) sin(\frac{x + a}{2})|\] Keep in mind that the sine function is always between -1 and 1, so we can apply the inequality \(-1 \leq sin(t) \leq 1\)

Step 4: Apply Inequalities for Sine Function and Combine them with Limit Definition

We apply inequality for sine function, and thus have \(-2 |sin(\frac{x - a}{2}) sin(\frac{x + a}{2})| \leq -2 sin(\frac{x - a}{2}) sin(\frac{x + a}{2}) \leq 2 |sin(\frac{x - a}{2}) sin(\frac{x + a}{2})|\) Now, for every ε > 0, there exists δ > 0, such that if 0 < |x - a| < δ, then \(-2 ε \leq -2 sin(\frac{x - a}{2}) sin(\frac{x + a}{2}) \leq 2 ε\) Since sin(x) is continuous and differentiable, we know that \(\lim _{x \rightarrow 0} sin(x) = 0\). This means that \[\lim _{x \rightarrow a} sin(\frac{x - a}{2}) sin(\frac{x + a}{2}) = 0\] As a result, we can choose δ > 0 such that \[ -2 sin(\frac{x - a}{2}) sin(\frac{x + a}{2}) \leq -2 ε\] and \[2 sin(\frac{x - a}{2}) sin(\frac{x + a}{2}) \leq 2 ε\] Since we have proven these inequalities for every ε, it proves that \[\lim_{x \rightarrow a} \cos x = \cos a\]

Key Concepts

Trigonometric Limits

The limit of a trigonometric function as a variable approaches a certain value concerns understanding how the function behaves close to that point. Trigonometric limits often involve functions like sine, cosine, and tangent. When considering limits at particular points, the goal is to determine the output value the function approaches.
For example, in the limit \[\lim _{x \to a} \cos x = \cos a,\]the task is to show that the cosine of \(x\), as \(x\) nears \(a\), approaches the cosine of \(a\).

• Trigonometric limits use algebraic manipulation and well-accepted trigonometric identities to simplify expressions.
• Evaluating these limits often requires understanding core properties of trigonometric functions, such as their periodic nature and bounded values.

Recognizing and using trigonometric identities simplifies these limit problems, making them more approachable.

Continuity

Continuity of a function at a particular point means there is no interruption in the graph of the function at that point. For a function to be continuous at a point \(a\), three conditions must be satisfied:

• \(f(a)\) must be defined.
• The limit \(\lim_{x \to a} f(x)\) must exist.
• \(\lim_{x \to a} f(x) = f(a)\).

In the context of trigonometric functions such as cosine, these functions are continuous everywhere because they smoothly pass through all points without jumps or breaks. This property is essential in proving limits because it allows us to assume that as \(x\) approaches \(a\), the function's value smoothly approaches \(\cos a\).
Continuity not only aids in simplifying limit calculations but also in understanding the behavior of functions across their domains.

Epsilon-Delta Definition

The epsilon-delta definition of a limit rigorously defines what it means for a function to approach a certain value as the input approaches a particular point. It asserts that for every given epsilon \(\varepsilon\) greater than zero, there exists a delta \(\delta\) such that whenever \(|x - a| < \delta\), it follows that \(|f(x) - L| < \varepsilon\).
This formalism is crucial for rigorously proving limits, especially when dealing with more complex functions like trigonometric ones.

• The epsilon represents how close the function's value should be to the limit \(L\).
• The delta represents how close \(x\) should be to \(a\) for \(f(x)\) to be that close to \(L\).

Employing the epsilon-delta definition requires a precise understanding of how to manipulate inequalities and often utilizes auxiliary functions or identities to bridge gaps between \(|x - a|\) and \(|f(x) - L|\). This method underpins the mathematical rigor necessary for proving limit properties.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all variables within their definitions. These identities are pivotal tools in trigonometry and calculus, particularly in the manipulation and simplification of expressions.
In the context of limits, identities like \[\cos(x) - \cos(a) = -2\sin\left(\frac{x - a}{2}\right)\sin\left(\frac{x + a}{2}\right)\]are employed to express differences in ways that are conducive to limit evaluation.

• Simplifying trigonometric expressions often requires recognizing opportunities to apply these identities.
• They allow translation of complex trigonometric expressions into simpler forms amenable to limit evaluation using algebraic tools like factorization and inequality.

By understanding and using these identities, you can bridge the gap between complex expressions and the simplified forms needed to evaluate and prove limits.