Problem 34 - Calculus for Biology and Medicine

Question

Use l'Hospital's rule to find the limits. $$ \lim _{x \rightarrow 1^{-}}(1-x) \tan \left(\frac{\pi}{2} x\right) $$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Identify the Indeterminate Form

We are given the limit \( \lim _{x \rightarrow 1^{-}}(1-x) \tan \left(\frac{\pi}{2} x\right) \). First, we recognize that as \( x \to 1^{-} \), both \( 1-x \to 0 \) and \( \tan \left(\frac{\pi}{2} x\right) \to \infty \). Therefore, this expression is of the indeterminate form "0 \( \cdot \) infinity". We should rearrange it to apply l'Hospital's Rule.

Step 2: Rewrite the Expression

To apply l'Hospital's Rule, we need this expression to be in a form of \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). Rewrite the expression as\[\lim _{x \rightarrow 1^{-}}\frac{(1-x)}{\cot \left( \frac{\pi}{2} x \right)},\]since \( \tan(x) = \frac{1}{\cot(x)} \).

Step 3: Apply l'Hospital's Rule

Now, we have the limit in the form \( \frac{0}{0} \), because \( 1-x \to 0 \) and \( \cot \left(\frac{\pi}{2} x\right) \to 0 \) as \( x \to 1^{-} \). Apply l'Hospital's Rule by differentiating the numerator and denominator:The derivative of \( 1-x \) is \( -1 \).The derivative of \( \cot \left( \frac{\pi}{2} x \right) \) is \(-\frac{\pi}{2} \csc^2 \left( \frac{\pi}{2} x \right)\).So, the limit becomes:\[\lim _{x \rightarrow 1^{-}}\frac{-1}{-\frac{\pi}{2} \csc^2 \left( \frac{\pi}{2} x \right)}\]

Step 4: Simplify the Expression

Simplify the expression:\[\lim _{x \rightarrow 1^{-}}\frac{-1}{-\frac{\pi}{2} \csc^2 \left( \frac{\pi}{2} x \right)} = \lim _{x \rightarrow 1^{-}}\frac{2}{\pi \csc^2 \left( \frac{\pi}{2} x \right)}\]Recall that \( \csc(x) = \frac{1}{\sin(x)} \), hence \( \csc^2 \left( \frac{\pi}{2} x \right) = \frac{1}{\sin^2 \left( \frac{\pi}{2} x \right)} \). Since \( \sin \left( \frac{\pi}{2} x \right) \to \sin(0)=0 \) as \( x \to 1^{-} \), the expression goes to \( 0 \).

Step 5: Compute Final Limit

Given that \( \csc^2 \left( \frac{\pi}{2} x \right) \to +\infty \) as \( x \to 1^{-} \), \( \frac{2}{\pi \csc^2 \left( \frac{\pi}{2} x \right)} \to 0 \). Thus, the limit is\[\lim _{x \rightarrow 1^{-}}(1-x) \tan \left( \frac{\pi}{2} x \right) = 0.\]

Key Concepts

Understanding Indeterminate Forms

Indeterminate forms are expressions that, at first glance, cannot be easily evaluated as they represent an undefined or ambiguous state. These forms often arise in calculus, particularly when dealing with limits. Common examples are \(0/0\), \(\infty/\infty\), and \(0 \cdot \infty\).
In the given problem, as \(x\) approaches 1 from the left, we have (1 - x) approaching 0 while \(\tan(\frac{\pi}{2} x)\) approaches infinity. This creates the \(0 \cdot \infty\) form, which seems undefined. To solve this, we rearrange the expression into a rational form like \(0/0\) or \(\infty/\infty\) that allows us to use l'Hospital's Rule. This rearrangement turns the product into a division, thus making the form explicitly indeterminate, paving the way for applying calculus techniques.

Mastering Trigonometric Limits

Trigonometric limits often involve angles approaching critical values like 0 or \(\pi/2\), which are sensitive points in trigonometric functions. In our exercise, the function \(\tan\left(\frac{\pi}{2} x\right)\) approaches infinity as \(x\) approaches 1 from the left, due to the angle shifting towards \(\pi/2\).

• The tangent function has vertical asymptotes at odd multiples of \(\pi/2\), thus leading to infinite values as they are approached from either side.
• Rewriting \(\tan\left(\frac{\pi}{2} x\right)\) as \(\frac{1}{\cot\left(\frac{\pi}{2} x\right)}\) allows us to transform a multiplicative indeterminate form into a division, which is necessary for using l'Hospital’s Rule.

By understanding how trigonometric functions behave near their limits, you can better predict and simplify limits via algebraic manipulation or calculus rules.

Approaching Calculus Problem-Solving

Solving calculus problems like the one discussed involves multiple strategies. The key is recognizing the form of the limit and knowing the right mathematical approaches:

• First, identify the indeterminate form present in the limit. Recognizing if it's \(0/0\), \(\infty/\infty\), or another type helps in deciding the right strategy.
• Utilize known calculus rules, such as l'Hospital's Rule, which applies to forms like \(0/0\) and \(\infty/\infty\), to differentiate the numerator and the denominator until the expression is no longer indeterminate.
• Ensure simplification at every step: After differentiating, simplify the expression to recognize limits that may result in standard values or zero.
• Apply relevant trigonometric identities, like expressing \(\tan\) in terms of \(\cot\) or \(\csc^2\), to aid in transformation and simplification.

Carefully following these steps can simplify even the most complex calculus problems, showcasing the beauty and utility of calculus in resolving seemingly difficult mathematical challenges.