Question

How do you find the limit of `(sqrtx)/ln(x+1)` as x approaches 0+?

Answer 1

`lim_(xrarr0^+) [(sqrtx)/(ln(x+1))] = color(blue)(oo`

Explanation:

Without using special rules...

To find the limit of the function as `x` approaches `0` from the right, we can plug in successively smaller positive numbers that get closer and closer to zero (i.e. `0.1`, `0.01`, `0.001`, etc.)

`0.1:" "` `(sqrt(0.1))/(ln(0.1+1)) ~~ ul(3.318`

`0.01:" "` `(sqrt(0.01))/(ln(0.01+1)) ~~ ul(10.050`

`0.001:" "` `(sqrt(0.001))/(ln(0.001+1)) ~~ ul(31.639`

`...`

`10^-15:" "` `(sqrt(10^-15))/(ln(10^-15+1)) ~~ ul(3.162xx10^7`

We can infer that the numbers get larger and larger as we go onward (which they do).

Also notice that in this limit, the denominator approaches `ln(1)`, which equals `0`, and as you divide a number by smaller and smaller numbers, the result approaches infinity, so we can conclude that

`lim_(xrarr0^+) [(sqrtx)/(ln(x+1))] = color(blue)(oo`