Evaluate the limit using algebraic techniques.
limit of (x^2 - 25)/(x^2 - 4x - 5) as x approaches 0.
How do I do this?
limit of (x^2 - 25)/(x^2 - 4x - 5) , as x --> 0
= lim(x+5)(x-5)/( (x-5)(x+1)
= lim (x+5)/(x+1) , as x ---> 0
= 5/1
= 5
To evaluate the limit of (x^2 - 25)/(x^2 - 4x - 5) as x approaches 0, you can use algebraic techniques. Here's how you can approach it:
Step 1: Simplify the expression
First, simplify the numerator and denominator. In this case, the numerator can be factored using the difference of squares formula, and the denominator can be factored using the quadratic formula or by factorization.
The numerator, x^2 - 25, factors as (x + 5)(x - 5).
The denominator, x^2 - 4x - 5, factors as (x - 5)(x + 1).
So the expression simplifies to (x + 5)(x - 5) / (x - 5)(x + 1).
Step 2: Cancel out common factors
Since there is a (x - 5) term present in both the numerator and the denominator, you can cancel it out.
Canceling out the common factors, the expression becomes (x + 5) / (x + 1).
Step 3: Evaluate the limit
Now that the expression is simplified, you can evaluate the limit as x approaches 0 by substituting x = 0 into the simplified expression.
Plugging in x = 0, you get (0 + 5) / (0 + 1) = 5 / 1 = 5.
Therefore, the limit of (x^2 - 25)/(x^2 - 4x - 5) as x approaches 0 is 5.