Understanding Limits: Breaking Down lim x→8 (1/x - 1/8)/(x - 8)

Let's examine a challenging limit problem that leads to an indeterminate form.When we try to evaluate this limit by directly substituting x equals 8, let's see what happens.We get zero divided by zero, which is an indeterminate form.But why exactly is zero divided by zero considered indeterminate? Let's understand this concept better.To visualize why we can't determine the limit through direct substitution, let's look at a coordinate plane.Different paths of approach to the point can lead to different limit values, which is why we need a different method to evaluate this limit.Since direct substitution leads to an indeterminate form, we'll need to transform this expression algebraically to find the true limit.To resolve this indeterminate form, we need to manipulate the expression algebraically.First, let's find a common denominator in the numerator. We'll convert one-over-x minus one-over-eight into a single fraction.Now we can substitute this simplified numerator back into our original limit expression.Next, we'll rewrite this as a multiplication of fractions, which will help us see the common factors.Notice that x minus 8 appears in both numerator and denominator. We can also rewrite 8 minus x as negative x minus 8.Now we can cancel the common factor of x minus 8 in both numerator and denominator.Now that we've manipulated our expression, let's simplify it to find our final answer.First, let's multiply by negative one over negative one, which equals one, to help with our cancellation.Notice that eight minus x in the numerator can be rewritten as negative x minus eight.After cancelling x minus eight in the numerator and denominator, our expression simplifies to negative one over eight x.Now that we have eliminated the troublesome x minus eight term, we can safely substitute x equals eight.This gives us our final answer: negative one over sixty-four.Let's verify that this answer makes sense in the context of our original limit problem.Our answer is a finite value, which is what we hoped to find. We've eliminated the zero over zero indeterminate form through algebraic manipulation. And the negative sign in our answer aligns with the behavior of the original expression as x approaches eight.