# How to Graph x/ln(x)

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

In this lesson, we explore how to graph the function x/ln x. We carefully explain how this function has a restricted domain and some fascinating features whose complete description involves limits and derivatives.

## What Can We Say About x/ln x ?

In this lesson, we show how to plot x/ln x. Let's introduce this plot by looking at the numerator and denominator separately.

In the numerator, we have simply x. Thus, all values of x are allowed and the plot of y = x is a straight line passing through the origin.

The denominator needs more attention. Logarithm is undefined for negative values of x. We will stay on the right side of the y-axis. Logarithm of 0 goes to -∞ and ln 1 = 0. Thus, we have to be careful at x = 1 because the denominator of x/ln x will equal zero.

Both x and ln x go to ∞ at x goes to ∞.

We have the ratio of two functions. At x = 0, 1 and ∞ the ratio is said to be indeterminate. Examples of indeterminate ratios are 0/0, 0/∞, 0/-∞, ∞/∞, -∞/∞ and ∞/-∞. In these cases, finding the limit by substitution will not work. However, the limit of the ratio is the limit of the ratio of the derivatives of each function. After differentiating, we can often substitute to find the limit. This is called l'Hospital's Rule and summarized as:

The numerator is f(x) and the denominator is g(x). We will use l'Hospital's rule and find the limit of the ratio of the derivatives.

## Plotting x/ln x

#### Step 1: Use l'Hospital's Rule to explore near x = 0.

At x = 0, we have 0/-∞. We use l'Hospital's rule.

The derivative of the numerator, x, is 1. The derivative of the denominator, ln x, is 1/x. Then,1 divided by 1/x is x. Thus, in the limit as x approaches 0, x/ln x is x. We can substitute x with 0 and get 0.

As x gets slightly larger than 0 but is still less than 1, the numerator is positive, but the denominator is negative. Thus, the curve of x/ln x goes negative as we move to the right of x = 0.

#### Step 2: Analyze what happens as x approaches 1 from the left.

Let's evaluate x/ln x for some values which get close to x = 1. For example, for x = .8, x/ln x is .8/ln .8 = -3.585… ≅ -3.6. Getting still closer to x =1 from the left, we choose x = .9. Then, x/ln x = .9/ln .9 = -8.542… = ≅ -8.5. See how getting closer and closer to x =1 from the left results in getting larger and larger negative values?

We could continue probing by picking values even closer to x = 1 and the result will be x/ln x approaches -∞ as x approaches 1 from the left.

#### Step 3: Analyze what happens as x approaches 1 from the right.

The reason we are careful about the direction of the approach now becomes clearer. Let's approach x = 1 from the right.

For x = 1.2, x/ln x = 1.2/ln 1.2 = 6.581… ≅ 6.58. The result is positive.

For an x value still closer to 1, like x = 1.1, we get 1.1/ln 1.1 = 11.541… ≅ 11.5. We are headed to positive ∞ as x approaches 1 from the right.

#### Step 4: Determine what happens as x gets very large.

As x → ∞, x/ln x goes to ∞/∞. Using l'Hospital's rule, the ratio, x/ln x becomes x. As x goes to ∞ we get ∞. Let's convince ourselves of this by taking larger and larger values of x.

To unlock this lesson you must be a Study.com Member.

### Register to view this lesson

Are you a student or a teacher?

#### See for yourself why 30 million people use Study.com

##### Become a Study.com member and start learning now.
Back
What teachers are saying about Study.com

### Earning College Credit

Did you know… We have over 200 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.