Question

Determine the infinite limit.

\(\mathop {\lim }\limits_{x \to {0^ + }} \,\,\left( {\frac{1}{x} - \ln x} \right)\)

Short answer

The limit tends to infinity.

Step-by-step solution

Analyze the given function at the given point

The given function has the argument of logarithmic function as\(x\).

As \(x\) approaches 0 from the right, then the argument also approaches 0 from the right. So, the logarithmic function must tend to negative infinity.

So, another term \(\frac{1}{x}\) approaches positive infinityas \(x\) approaches 0 from the right. Thus, \(\frac{1}{x} \to \infty \), and \(\ln x \to - \infty \) as \(x \to {0^ + }\).

Estimate the limit

The final limit is calculated as subtraction of negative infinity from positive infinity, which results in positive infinity, as shown below:

\(\begin{aligned}\mathop {\lim }\limits_{x \to {0^ + }} \left( {\frac{1}{x} - \ln x} \right) &= \infty - \left( { - \infty } \right)\\ &= \infty \end{aligned}\)

Thus, the limit tends to infinity.