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 of the limit

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 the numerator and the denominator

Differentiate the numerator: \((5x - 35)'=5\)
Differentiate the denominator: \((5x^{2}-245)'=10x\)
So, \(\lim_{x
ightarrow7}\frac{5x - 35}{5x^{2}-245}=\lim_{x
ightarrow7}\frac{5}{10x}\)

Step3: Evaluate the new limit

Substitute \(x = 7\) into \(\frac{5}{10x}\), we get \(\frac{5}{10\times7}=\frac{1}{14}\)

Answer:

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