Question

use lhôpitals rule to find the limit.

lim 5x - 35
x→7 5x² - 245

lim 5x - 35
x→7 5x² - 245 =

Explanation:

Step1: Check the form

When \(x = 7\), the numerator \(5x-35=5\times7 - 35=0\), and the denominator \(5x^{2}-245=5\times7^{2}-245 = 5\times49-245=0\). So, it is in the \(\frac{0}{0}\) form, and we can apply L'Hôpital's Rule.

Step2: Differentiate numerator and denominator

Differentiate the numerator: \((5x - 35)'=5\).
Differentiate the denominator: \((5x^{2}-245)'=10x\).

Step3: Find the limit

Now, we find \(\lim_{x
ightarrow7}\frac{5}{10x}\). Substitute \(x = 7\) into \(\frac{5}{10x}\), we get \(\frac{5}{10\times7}=\frac{1}{14}\).

Answer:

\(\frac{1}{14}\)