lim x->3 ((x^-2)-(3^-2))/(x-3)
I see it as
lim [1/x^2 - 1/9]/(x-3) as x-->3
using a common denominator of 9x^2 and adding up the square bracket, we get
lim[(9-x^2)/(9x^2)]/(x-3)
= lim[(3-x)(3+x)/9x^2]/(x-3)
= lim [-(3+x)/9x2] as x --> 3
= -6/81 = -2/27
Wow, this was complicated. Had to draw everything out and look at it a few times. Thank you very much! :D
To evaluate the limit of the given expression, we need to simplify it first. Let's break it down step by step:
Step 1: Simplify the numerator.
The numerator is ((x^-2) - (3^-2)). We can use the formula a^-n = 1 / (a^n) to simplify it. Applying this formula, we get:
((x^-2) - (3^-2)) = (1 / x^2) - (1 / 3^2) = (1 / x^2) - (1 / 9)
Step 2: Simplify the denominator.
The denominator is (x - 3), which is already simplified.
Step 3: Rewrite the expression.
Now that we have simplified the numerator and the denominator, we can rewrite the given expression:
((x^-2) - (3^-2))/(x - 3) = ((1 / x^2) - (1 / 9))/(x - 3)
Step 4: Factorize the numerator.
To make further progress, we can factorize the numerator by finding a common denominator:
((1 / x^2) - (1 / 9))/(x - 3) = (9 - x^2) / (9 * x^2)
Step 5: Apply the limit.
Now, we can apply the limit as x approaches 3. Substituting 3 into the expression, we get:
lim x->3 ((1 / x^2) - (1 / 9))/(x - 3) = (9 - 3^2) / (9 * 3^2) = (9 - 9) / (81)
Simplifying further, we get:
lim x->3 ((x^-2) - (3^-2))/(x - 3) = 0 / 81 = 0
Therefore, the limit of the given expression as x approaches 3 is 0.