Question

Use l’Hôpital’s Rule to find the limits in Exercises 85–108.104. lim(x→4) (sin²(πx))/(e^(x-4) + 3 - x)

Accepted Answer

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. 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}$$(\pi x) as$$ $$2 \sin(\pi x) \cdot \cos(\pi x) \cdot \pi. $$For the denominator, differentiate $$e^{x-4} + 3 - x$ with respect to x$. The derivative of e$$^{x-4}$$ is e$$^{x-4}$$, the derivative of 3$ is 0$, and the derivative of -x$ is -1$. Write the new limit expression as $$\lim_{x 	o 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. 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.

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 video answer for a similar problem:

Key Concepts

l’Hôpital’s Rule

Evaluating Limits Involving Trigonometric Functions

Exponential and Polynomial Functions in Limits