what is the limit as x -> 0+ (0 from the right) of |x|/x?
from the right, x/x = 1
To find the limit as x approaches 0 from the positive side of |x|/x, we need to consider the behavior of the function as x gets closer to 0 from the right.
When x is positive and approaches zero, |x| is equal to x since x is positive. So, |x|/x simplifies to x/x, which equals 1 for x ≠ 0.
Therefore, as x approaches 0+ (0 from the right), the limit of |x|/x is 1.
We can also verify this result using the formal definition of limits. Let's explain the process:
1. Start by defining the given function: f(x) = |x|/x.
2. Determine the limit as x approaches 0+ of f(x), denoted as lim(x→0+) f(x).
3. Considering that x > 0 as x approaches 0, we can simplify the function to f(x) = x/x for x ≠ 0.
4. Now, substitute 0 into the simplified function: lim(x→0+) (x/x) = 0/0, but this is an indeterminate form.
To proceed, we can use L'Hôpital's rule, which states that if the limit of the quotient of two functions is of the form 0/0 or ∞/∞, then it is equal to the limit of the quotient of their derivatives.
5. Take the derivative of the numerator and denominator: f'(x) = 1/1 = 1.
6. Evaluate the limit of the derivative: lim(x→0+) 1 = 1.
7. Since the limit of the derivative equals 1, we can conclude that lim(x→0+) (x/x) = 1.
Therefore, the limit as x approaches 0+ of |x|/x is indeed equal to 1.