Hello! If you're struggling on Bezros' storygame 'Night Shift' I gotcha. On one of the branches, you'll get an equation that looks like this:
lim [x?0] [(e^2x)-1]/x.
Some of you may have never seen a problem like this in your life and just picked until you got the right answer. But in this thread, I'll show you how it's done.
So, this question is asking you to find the limit as x approaches 0. That means plugging in 0 into the equation. I will solve the equation arithmetically. Just note that e is it's own number, equaling 2.71828... Also, the '^' symbol is an exponent power.
Step 1: {[e^2(0)]-1}/(0)
Step 2: [(e^0)-1]/0
Step 3: (1-1)/0
Step 4: 0/0
What results is called the Indeterminate Form, or 0/0. You can't divide 0 by 0, so it creates a paradox. However, there is a way to solve this.
However, in order to solve this properly, we first need a knowledge of derivatives, which is basically a function for the slope of a function. To learn more, you can read this article to get a head start.
Now, assuming what derivatives are and how to find them, we will introduce the main method to solve this problem: L'Hôpital's Rule. Basically, if you're finding a limit and the result is an indeterminate form, you can differentiate both the numerator and denominator functions to get a proper answer.
For the function mentioned, we will differentiate (e^2x)-1 and x respectively. Here's the end result after applying L'Hop's Rule:
[2e^(2x)]/1
Now we can solve for the limit as x approaches 0 without getting the indeterminate form.
Step 1: Remove the /1. The function simplifies to 2e^(2x)
Step 2: 2e^[2(0)]
Step 3: 2e^(0)
Step 4: 2(1)
Step 5 2
Thus, the limit as x approaches 0 is 2.
