If a ≠ 0, then the limit of (x^2 - a^2)/(x^4 - a^4) as x approaches a is:
How do I do this type of problem?
A simple case of factoring will do the job
limit (x^2 - a^2)/(x^4 - a^4) , x ---> a
= lim (x^2 - a^2)/( (x^2 + a^2)(x^2 - a^2)
= lim 1/(x^2 + a^2) , as x --> a
= 1/(2a^2)
How did you simplify 1/(x^2 + a^2) as 1/(2a^2)?
Never mind, I got it.
Well, let's approach this problem with a clownish twist! Imagine you are at a circus, trying to calculate this limit. You see a strongman juggling elephants while standing on a tightrope. That's quite a sight, right?
Now, let's focus on the limit itself. We see that both the numerator and denominator are expressions involving powers of x, with a little twist of the number a thrown in. It's like trying to juggle different objects, but with the added challenge of an elephant in the mix!
To simplify this problem and avoid getting overwhelmed like that strongman, we can factor both the numerator and denominator. The numerator can be factored as (x - a)(x + a), and the denominator can be factored as (x^2 - a^2)(x^2 + a^2).
Now, cancel out the common terms of (x - a) and (x + a). We are left with 1/(x^2 + a^2).
Now, here's where the magic happens. As x approaches a, the term (x - a) shrinks to zero, making the denominator (x^2 + a^2) become non-zero. Since the denominator will never be zero, the final answer to our circus-like limit is 1/(a^2 + a^2) which simplifies to 1/(2a^2).
So, in clown terms, the limit of (x^2 - a^2)/(x^4 - a^4) as x approaches a is 1/(2a^2). Ta-da! Keep practicing these circus-style problems, and you'll become a master acrobat of limits in no time!
To find the limit of (x^2 - a^2)/(x^4 - a^4) as x approaches a, you can simplify the expression and then substitute the value of x = a into the simplified expression. Here are the steps:
Step 1: Factor the numerator and denominator. Since (a^2 - b^2) = (a + b)(a - b) and (a^4 - b^4) = (a^2 + b^2)(a^2 - b^2), we can factor the expression as follows:
(x^2 - a^2)/(x^4 - a^4) = [(x - a)(x + a)]/[(x^2 + a^2)(x^2 - a^2)].
Step 2: Cancel out the common factors. The (x - a) and (x + a) in the numerator and the (x^2 - a^2) in the denominator can be canceled out, leaving:
(x - a)/(x^2 + a^2).
Step 3: Substitute x = a. Plug in x = a into the simplified expression:
(a - a)/(a^2 + a^2) = 0/(2a^2) = 0.
Therefore, the limit of (x^2 - a^2)/(x^4 - a^4) as x approaches a is 0.
To find the limit of the given expression as x approaches a, we can use algebraic manipulation and factorization. Here's how you can approach this problem step by step:
1. Start by factoring the numerator and the denominator. This will help us simplify the expression.
The numerator, x^2 - a^2, can be factored as (x - a)(x + a).
The denominator, x^4 - a^4, can be factored as (x^2 - a^2)(x^2 + a^2).
2. Now we can cancel out the common factor, which is (x - a) and (x + a) in the numerator and denominator:
(x - a)(x + a) / (x^2 - a^2)(x^2 + a^2).
3. Next, notice that (x + a) cancels out with (x^2 + a^2) in the denominator, resulting in:
(x - a) / (x^2 - a^2).
4. At this point, we can see that (x - a) also appears in the denominator. We can cancel out this factor as well:
1 / (x + a).
5. Finally, as x approaches a, the expression can be evaluated:
As x approaches a, (x + a) approaches 2a, so the limit becomes:
1 / (2a).
So, the limit of (x^2 - a^2)/(x^4 - a^4) as x approaches a is 1 / (2a).