Evaluate the following limits: (i) Lt x→−1 (x²−1)/(x+1)
Understand the Problem
The question is asking to evaluate a limit expression involving a rational function as x approaches a specific value. The high-level approach involves substituting the value of x into the function and simplifying to find the limit.
Answer
The limit is $-2$.
Answer for screen readers
The limit is ( -2 ).
Steps to Solve
- Substitute ( x = -1 ) in the expression
Start by substituting ( x = -1 ) into the limit expression: $$ \frac{(-1)^2 - 1}{-1 + 1} $$ This evaluates to: $$ \frac{0}{0} $$ This is an indeterminate form.
- Factor the numerator
To resolve the indeterminate form, we need to factor the numerator ( x^2 - 1 ): $$ x^2 - 1 = (x - 1)(x + 1) $$
- Rewrite the limit expression
Substituting the factored form into the limit expression, we get: $$ \lim_{x \to -1} \frac{(x - 1)(x + 1)}{x + 1} $$ Notice that ( x + 1 ) cancels out: $$ \lim_{x \to -1} (x - 1) $$
- Evaluate the simplified limit
Now substitute ( x = -1 ) in the simplified expression: $$ -1 - 1 = -2 $$
The limit is ( -2 ).
More Information
This limit was evaluated by recognizing that direct substitution led to an indeterminate form. Factoring allowed us to simplify the expression, making it easier to find the limit.
Tips
- Failing to recognize ( \frac{0}{0} ) as an indeterminate form and proceeding to calculate the limit without simplifying.
- Forgetting to factor the numerator, which is essential when confronted with indeterminate forms.
