What is the limit: lim (x -> 0) (e^x - 1)/x? (2022)
Practice Questions
Q1
What is the limit: lim (x -> 0) (e^x - 1)/x? (2022)
1
0
e
Undefined
Questions & Step-by-Step Solutions
What is the limit: lim (x -> 0) (e^x - 1)/x? (2022)
Step 1: Understand the limit we want to find: lim (x -> 0) (e^x - 1)/x.
Step 2: Recognize that e^x is a function that approaches 1 when x is 0.
Step 3: Substitute x = 0 into the expression (e^x - 1)/x. This gives us (e^0 - 1)/0, which is 0/0, an indeterminate form.
Step 4: To resolve the indeterminate form, we can use L'Hôpital's Rule, which states that if we have 0/0, we can take the derivative of the top and the bottom.
Step 5: The derivative of the top (e^x - 1) is e^x, and the derivative of the bottom (x) is 1.
Step 6: Now we can rewrite the limit as lim (x -> 0) e^x/1.
Step 7: Substitute x = 0 into e^x, which gives us e^0 = 1.
