Question

Evaluate the following limits or state that they do not exist. $$\lim _{x \rightarrow \pi / 2} \frac{\cos x}{x-(\pi / 2)}$$

Short answer

Answer: The value of the limit is -1.

Step-by-step solution

Identify the indeterminate form form

Before applying L'Hospital's rule, we must confirm that the given limit is in the indeterminate form of \(0/0\). As x approaches π/2, we have: $$\lim_{x \rightarrow \pi / 2} \cos x = \cos (\pi / 2) = 0$$ and $$\lim_{x \rightarrow \pi / 2} (x - (\pi / 2)) = 0$$ Since both the numerator and the denominator tends to zero, the given limit is in the indeterminate form \(\frac{0}{0}\).

Apply L'Hospital's rule

Apply L'Hospital's rule by taking the derivatives of the numerator and the denominator with respect to x, and then find the limit of their quotient. The derivative of the numerator with respect to x: $$\frac{d(\cos x)}{dx} = -\sin x$$ The derivative of the denominator with respect to x: $$\frac{d(x - (\pi / 2))}{dx} = 1$$ Now calculate the limit of the quotient of the derivatives. $$\lim_{x \rightarrow \pi / 2} \frac{-\sin x}{1} = -\sin(\pi / 2) = -1$$ The limit of the given expression as x approaches π/2 is -1. The answer is: $$\lim _{x \rightarrow \pi / 2} \frac{\cos x}{x-(\pi / 2)} = -1$$

Key concepts

L'Hospital's Rule

L'Hospital's rule is a widely used technique in calculus for evaluating limits that yield indeterminate forms. The rule states that if you have a limit of the form \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \) and the functions involved are differentiable, then the limit of the ratios of their derivatives may provide the answer.

To apply L'Hospital's rule, you first differentiate the numerator and the denominator separately and then take the limit of the new fraction. If the resulting limit is still indeterminate, you might have to apply the rule again until a conclusive result is obtained or until it becomes apparent that the limit does not exist.

Importance of Checking Conditions

It's crucial to verify that the original limit produces an indeterminate form and that the functions are differentiable within the interval around the point of interest. Only then can L'Hospital's rule be appropriately utilized. This step ensures that the conclusions drawn are mathematically sound.

Indeterminate Forms

Indeterminate forms present a challenge in calculus because they do not immediately reveal the behavior of a function as it approaches a certain point. The most common indeterminate forms are \( \frac{0}{0} \) and \( \frac{\infty}{\infty} \) but there are others, such as \(0 \cdot \infty\), \(\infty - \infty\), \(0^0\), \(1^\infty\), and \(\infty^0\).

These forms are deemed indeterminate because the rules of arithmetic do not provide a way to discern a unique limit. Tools like L'Hospital's rule help to resolve these ambiguities by working with derivatives, which can simplify expressions and make limits easier to evaluate.

Understanding Limits Through Behavior

Recognizing indeterminate forms is just the initial step. The deeper goal is to understand the behavior of functions near the points that produce these forms and to use appropriate methods, like L'Hospital's rule, to uncover the true nature of the limits.

Derivatives

Derivatives are at the heart of calculus, serving as a fundamental tool for understanding the rate at which a function changes with respect to its input. They can predict the slope of a tangent to a curve at a given point, indicating the direction and steepness of the curve.

Mathematically, the derivative of a function at a certain point is the limit of the ratio of the change in the function's value to the change in the input value, as the change in the input approaches zero. It's symbolically represented as \(\frac{df}{dx}\) when the function is f and the input is x.

Role in L'Hospital's Rule

The application of derivatives is crucial when using L'Hospital's rule. As seen in the provided textbook solution, the derivatives of the numerator and the denominator function are used to resolve the indeterminate form by finding the limit of their ratio. This transforms a potentially complex calculation into a more manageable one, often revealing the behavior of a function at points where direct substitution is ineffective.