Questions: If the following limit is of indeterminate form, use l'Hôpital's Rule to find it. If the limit is not of indeterminate form, find it by other means. lim x -> 0+ 9x ln(7x)

Solution Steps

We need to evaluate the limit \[ \lim _{x \rightarrow 0^{+}} 9 x \ln (7 x). \] As \(x \rightarrow 0^{+}\), the expression \(x \ln(7x)\) approaches the indeterminate form \(0 \cdot (-\infty)\).

To apply l'Hôpital's Rule, we rewrite the limit in a suitable form: \[ \lim _{x \rightarrow 0^{+}} 9 x \ln(7 x) = \lim _{x \rightarrow 0^{+}} \frac{\ln(7x)}{1/x}. \] This transforms our limit into the form \(\frac{-\infty}{\infty}\), which allows us to use l'Hôpital's Rule.

We differentiate the numerator and the denominator:

• The derivative of \(\ln(7x)\) is \(\frac{1}{7x} \cdot 7 = \frac{1}{x}\).
• The derivative of \(1/x\) is \(-\frac{1}{x^2}\).

Applying l'Hôpital's Rule gives us: \[ \lim _{x \rightarrow 0^{+}} \frac{\frac{1}{x}}{-\frac{1}{x^2}} = \lim _{x \rightarrow 0^{+}} -x = 0. \]

Final Answer