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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.


Plot of y = x
f(x)=x


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.


Plot of ln x
g(x)=ln_x


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:


l


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.


l


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.


Near x = 0
near_x=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?


Near x = 1 from the left
near_x=1_from_the_left


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.


Near x = 1 from the right
near_x=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.

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