Final answer:
The limit as x approaches 5 of (x-5)/(x²-25) is 1/10. Factor the denominator and cancel the common factor of (x-5) in the numerator and denominator to simplify the expression to 1/(x+5), then evaluate the limit by substitution.
Step-by-step explanation:
To evaluate the limit of the function (x-5)/(x²-25) as x approaches 5, we first need to recognize that direct substitution of x = 5 into the function gives us a 0/0 indeterminate form. This suggests that we should simplify the expression before evaluating the limit.
We can factor the denominator as a difference of squares, so x² - 25 can be rewritten as (x - 5)(x + 5). Now we see that the numerator x - 5 is a common factor in both the numerator and denominator. Thus, we can simplify the expression by canceling out the common factor.
The simplified function is then 1/(x + 5). Now, we can safely substitute x = 5 into this simplified function without causing an undefined expression, giving us 1/(5 + 5) = 1/10. Therefore, the limit as x approaches 5 of (x-5)/(x²-25) is 1/10.