Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
104. lim(x→4) (sin²(πx))/(e^(x-4) + 3 - x)

Verified step by step guidance

1. First, identify the form of the limit by substituting \(x = 4\) into the expression \(\frac{\sin^{2}(\pi x)}{e^{x-4} + 3 - x}\). Check if it results in an indeterminate form like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), which allows the use of l'Hôpital's Rule.
2. If the limit is an indeterminate form, apply l'Hôpital's Rule by differentiating the numerator and denominator separately with respect to \(x\). For the numerator, use the chain rule to differentiate \(\sin^{2}(\pi x)\) as \(2 \sin(\pi x) \cdot \cos(\pi x) \cdot \pi\).
3. For the denominator, differentiate \(e^{x-4} + 3 - x\) with respect to \(x\). The derivative of \(e^{x-4}\) is \(e^{x-4}\), the derivative of \(3\) is \(0\), and the derivative of \(-x\) is \(-1\).
4. Write the new limit expression as \(\lim_{x \to 4} \frac{2 \sin(\pi x) \cos(\pi x) \pi}{e^{x-4} - 1}\) and then substitute \(x = 4\) into this expression to evaluate the limit.
5. If the resulting expression is still an indeterminate form, repeat the differentiation process using l'Hôpital's Rule again until the limit can be evaluated directly.

Key Concepts

l’Hôpital’s Rule

l’Hôpital’s Rule is a method for evaluating limits that result in indeterminate forms like 0/0 or ∞/∞. It states that the limit of a ratio of functions can be found by taking the limit of the ratio of their derivatives, provided certain conditions are met.

Evaluating Limits Involving Trigonometric Functions

When limits involve trigonometric functions like sine, it is important to understand their behavior near specific points. For example, knowing that sin(πx) approaches zero as x approaches 4 helps identify indeterminate forms and apply appropriate techniques.

Exponential and Polynomial Functions in Limits

Understanding how exponential functions like e^(x-4) and polynomial expressions behave near a point is crucial. This helps determine the form of the limit and whether l’Hôpital’s Rule applies, especially when the denominator approaches zero or a finite value.