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Limit of (x^2 - 1) / (x - 1) as x approaches 1
Published on October 19, 2024
Calculus
The limit of (x² - 1) / (x - 1) as x approaches 1 is 2. This is found by factoring the numerator (difference of squares) and canceling the common factor of (x - 1).
Question
Find the limit of f(x) = (x² - 1) / (x - 1) as x approaches 1
Answer
Let’s find the limit of the function f(x) = (x² - 1) / (x - 1) as x approaches 1.
1. Direct Substitution:
First, we try directly substituting x = 1 into the function:
f(1) = (1² - 1) / (1 - 1) = 0/0
This results in an indeterminate form (0/0), which means we need to use a different approach.
2. Factoring and Simplification:
Notice that the numerator is a difference of squares:
x² - 1 = (x + 1)(x - 1)
Now we can simplify the function:
f(x) = [(x + 1)(x - 1)] / (x - 1)
Since x approaches 1 but doesn’t actually equal 1, we can cancel the (x - 1) terms:
f(x) = x + 1
3. Limit Evaluation:
Now we can directly substitute x = 1 into the simplified function:
lim_(x→1) (x + 1) = 1 + 1 = 2
Therefore, the limit of f(x) = (x² - 1) / (x - 1) as x approaches 1 is 2.