Interactive Mathematics by Kunio Mitsuma, Ph.D.

KU Interactive Mathematics

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Limit Computation

The following problem reviews some important skills from precalculus. So, let’s take a look:

Computation of this limit starts with analysis of a rational function, then uses the graph of the natural logarithmic function.

First, by Laws of Logarithms, we obtain

Now, as x approaches ∞, we will first investigate where the blue goes.
Each of the three small fractions 3/x, 1/x², and another 1/x² will approach 0. And, since x approaches +∞, we also see that each of these small fractions remains positive. Symbolically, we could describe this analysis as:

Let

Then, when x approaches +∞, we see that u approaches 0⁺.
From the graph of v = ln(u) below, we know that v now approaches –∞. This is the answer.

As we see in the graph, as the input approaches 0⁺, the output of the ln function approaches –∞.