Question

Find the following limits or state that they do not exist. Assume \(a, b, c,\) and k are fixed real numbers. $$\lim _{x \rightarrow 0} \frac{\cos x-1}{\cos ^{2} x-1}$$

Short answer

Question: Evaluate the following limit: $$\lim _{x \rightarrow 0} \frac{\cos x-1}{\cos ^{2} x-1}$$ Answer: $$\frac{1}{2}$$

Step-by-step solution

Identify Possible Simplifications

First, notice that the denominator can be factored using the difference of squares formula: $$\cos ^2 x - 1 = (\cos x - 1)(\cos x + 1)$$ Now, let's rewrite the given limit using this factored form of the denominator: $$\lim _{x \rightarrow 0} \frac{\cos x-1}{(\cos x - 1)(\cos x + 1)}$$

Simplify the Limit Expression

Since we are taking the limit as \(x\) approaches \(0\), we know that the expression \(\cos x - 1\) will become \(0\). We can simplify the given expression by canceling out the common factor in the numerator and the denominator: $$\lim _{x \rightarrow 0} \frac{1}{\cos x + 1}$$

Evaluate the Limit

Now, we can evaluate the limit directly: $$\lim _{x \rightarrow 0} \frac{1}{\cos x + 1} = \frac{1}{\cos 0 + 1} = \frac{1}{1 + 1} = \frac{1}{2}$$ Thus, the limit we were looking for is: $$\lim _{x \rightarrow 0} \frac{\cos x-1}{\cos ^{2} x-1} = \frac{1}{2}$$

Key concepts

Limits

The concept of a limit is fundamental in calculus and describes how a function behaves as the input approaches a specific value. More formally, the limit of a function \(f(x)\) as \(x\) approaches a certain value \(c\) is the value that \(f(x)\) gets closer to as \(x\) gets closer to \(c\). If this value exists and is finite, we say the limit of \(f(x)\) as \(x\to c\) is equal to \(L\), and it is written as \(\lim_{x \to c} f(x) = L\). If the function does not approach any particular value, the limit is said not to exist.

Understanding limits helps us analyze the behavior of functions for points where they might not be well-defined. Here are some key aspects to consider:

• Evaluating limits directly: In many cases, you can plug in the value that \(x\) approaches directly into the function.
• Simplifying expressions: Sometimes algebraic manipulation is needed to resolve indeterminate forms like \(\frac{0}{0}\).
• Special limit rules: There are certain limits, especially involving trigonometric functions, that are well-known and useful to solve limits directly without extensive computation.

Trigonometric Limits

Trigonometric limits involve trigonometric functions like sine and cosine and often deal with angles tending to values where these functions reach their extremities, such as 0 or \(\frac{\pi}{2}\).

For many trigonometric limits, direct substitution could initially lead to indeterminate forms such as \(\frac{0}{0}\), which require simplification or using specific limit properties.

One of the most common techniques for handling trigonometric limits involves identities and limit properties:

• Standard limits: There are essential standard results that one should memorize for quick application:
  \(\lim_{x \to 0} \frac{\sin x}{x} = 1\) and \(\lim_{x \to 0} \frac{1 - \cos x}{x} = 0\).
• Substitution and simplification: Often, by substituting known values or simplifying trigonometric expressions, you can convert the expression into a form where these standard results apply.
• Factoring: Like in the original exercise where factoring the difference of squares was key, similar algebraic techniques help simplify complex trigonometric limits.

Limit Evaluation Steps

Solving limit problems often involves a step-by-step approach that simplifies the expression to resolve potential indeterminacies and enable direct evaluation.

The following steps illustrate a structured approach to solving limits, like in our example:

• Identify possible simplifications: Look for algebraic manipulations, such as factoring or expanding, that can simplify the expression.
• Simplify the expression: Remove terms that lead to the indeterminate form using cancellations, as seen by cancelling out \(\cos x - 1\) in the original solution.
• Direct evaluation: Once simplified, substitute the limiting value directly into the remaining expression to find the limit.
• Check the context: Confirm that your solution matches the context of the problem, ensuring that the approaches used respected the domain and properties of the function.

This step-by-step structure is not only useful but often necessary, as it ensures a formal and accurate calculation of limits. By applying these methods diligently, you can tackle even complex limit problems with confidence.