Question

Determine the infinite limit.

\(\mathop {\lim }\limits_{x \to 0} \,\,\left( {\ln {x^2} - {x^{ - 2}}} \right)\)

Short answer

The limit tends to negative infinity.

Step-by-step solution

Analyze the given function at the given point

The given function has the argument oflogarithmic function as\({x^2}\).

As\(x\)approaches 0, then theargumentapproaches 0. So, the logarithmic function must tend to negative infinity.

So, another term \({x^{ - 2}}\) approaches infinity as \(x\) approaches 0.

Estimate the limit

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

\(\begin{aligned}\mathop {\lim }\limits_{x \to 0} \left( {\ln {x^2} - {x^{ - 2}}} \right) &= - \infty - \infty \\ &= - \infty \end{aligned}\)

Thus, the limit tends to negative infinity.