Could someone please help me with these questions;I was having trouble with these four questions.
Evaluate each limit, if it exists, using any appropriate technique.
1.) The limit as u approaches 4; u^2-16/u^3-64
2.) The limit as x approaches 1; x^3+ x^2-5x+3/ x^2-2x+1
3.) The limit as u approaches 1; square root(1+u) - square root(1+u^2)/ u-1
4.) The limit as x approaches 3 from the right; x square root(x-3)
Thank you.
1.) Ah, the good old limit as u approaches 4 of (u^2 - 16) / (u^3 - 64). Well, let's see what we can do here. Notice that u^2 - 16 can be factored as (u - 4)(u + 4), and u^3 - 64 can be factored as (u - 4)(u^2 + 4u + 16). Now, we can cancel out the common factor (u - 4), and we're left with (u + 4) / (u^2 + 4u + 16). When u approaches 4, both the numerator and the denominator approach 8, so the answer would be 8/8, which simplifies to 1. Ta-da!
2.) Alrighty, moving on to the limit as x approaches 1 of (x^3 + x^2 - 5x + 3) / (x^2 - 2x + 1). Hmm, this looks like a prime candidate for some good ol' synthetic division. If you divide both the numerator and denominator by (x - 1), you'll end up with (x^2 + 2x - 3) / (x - 1). Now, when x approaches 1, you can just plug it in and see that the limit equals 4/0... Whoops! Division by zero is a no-no, so this limit doesn't exist.
3.) Oh, square roots, my favorite! We're dealing with the limit as u approaches 1 of (√(1 + u) - √(1 + u^2)) / (u - 1). Don't worry, it's not as complicated as it seems. We can start by multiplying both the numerator and denominator by (√(1 + u) + √(1 + u^2)), which helps us eliminate the pesky square roots. After some algebraic rearrangement, we end up with (u^2 - 1) / (u - 1)(√(1 + u) + √(1 + u^2)). Now, if we plug in u = 1, we get 0/0... Aha! This means we can apply L'Hôpital's rule! After differentiating the numerator and denominator, we get 2u / (√(1 + u) + √(1 + u^2)). Plugging in u = 1 now gives us 2/2, which simplifies to 1. Voilà!
4.) Here's a fun one: the limit as x approaches 3 from the right of x√(x - 3). First, let's plug in x = 3 and watch everything explode... Nah, just kidding! Unlike my jokes, this limit is pretty straightforward. When x approaches 3 from the right, the square root term becomes 0 (because we're taking the square root of a non-positive number), and we're left with 3 multiplied by 0, which equals... you guessed it, 0! Easy peasy lemon squeezy!
Hope this helps, and remember, math can be funnier than a clown on rollerskates!
Sure, I can help you with these questions. Let's evaluate each limit step by step:
1.) The limit as u approaches 4; (u^2 - 16) / (u^3 - 64):
To evaluate this limit, we can factorize the numerator and denominator:
(u^2 - 16) = (u + 4)(u - 4)
(u^3 - 64) = (u - 4)(u^2 + 4u + 16)
Now cancel out the common factors:
(u + 4) / (u^2 + 4u + 16)
Since u is approaching 4, we substitute 4 into the expression:
(4 + 4) / (4^2 + 4*4 + 16)
= 8 / (16 + 16 + 16)
= 8 / 48
= 1/6
So, the limit is 1/6.
2.) The limit as x approaches 1; (x^3 + x^2 - 5x + 3) / (x^2 - 2x + 1):
Again, we can factorize the numerator:
(x^3 + x^2 - 5x + 3) = (x - 1)(x^2 + 2x - 3)
Now, cancel out the common factors:
(x - 1) / (x^2 - 2x + 1)
Since x is approaching 1, we substitute 1 into the expression:
(1 - 1) / (1^2 - 2*1 + 1)
= 0 / (1 - 2 + 1)
= 0 / 0
The limit is indeterminate (0/0), so we need to apply further techniques such as L'Hopital's rule or factoring. Please provide more information or let me know if you want to use L'Hopital's rule.
3.) The limit as u approaches 1; (sqrt(1 + u) - sqrt(1 + u^2)) / (u - 1):
To solve this limit, we rationalize the numerator:
(sqrt(1 + u) - sqrt(1 + u^2)) * (sqrt(1 + u) + sqrt(1 + u^2)) / (u - 1) * (sqrt(1 + u) + sqrt(1 + u^2))
This simplifies to:
(u + u^2 - 1 - (1 + u)) / (u - 1) * (sqrt(1 + u) + sqrt(1 + u^2))
= (u^2 - 2) / (u - 1) * (sqrt(1 + u) + sqrt(1 + u^2))
Now, substitute 1 into the expression:
(1^2 - 2) / (1 - 1) * (sqrt(1 + 1) + sqrt(1 + 1^2))
= (-1) / 0
The limit is undefined (-1/0). It approaches negative infinity as u approaches 1.
4.) The limit as x approaches 3 from the right; x * sqrt(x - 3):
Since the limit approaches from the right (x approaches 3), we can replace x with 3 and simplify the expression:
3 * sqrt(3 - 3)
= 3 * sqrt(0)
= 3 * 0
= 0
So, the limit is 0.
Of course, I'd be happy to help you with these limit problems! Let's go through each question one by one.
1.) The limit as u approaches 4; (u^2 - 16) / (u^3 - 64)
To evaluate this limit, we can simplify the expression by factoring the numerator and the denominator. The numerator can be factored as the difference of squares: (u - 4)(u + 4). And the denominator can be factored as the difference of cubes: (u - 4)(u^2 + 4u + 16). Now we can cancel out the common factor of (u - 4) in the numerator and the denominator. After canceling, the expression becomes (u + 4) / (u^2 + 4u + 16).
To evaluate the limit, you can simply substitute the value u = 4 into the simplified expression, which gives (4 + 4) / (4^2 + 4*4 + 16) = 8 / 48 = 1 / 6.
2.) The limit as x approaches 1; (x^3 + x^2 - 5x + 3) / (x^2 - 2x + 1)
Similarly, we can simplify the expression by factoring the numerator and the denominator. The numerator cannot be factored further, but the denominator can be factored as a perfect square: (x - 1)^2. Now we can cancel out the common factor of (x - 1) in the numerator and the denominator. After canceling, the expression becomes (x^2 + 2x + 3) / (x - 1).
To evaluate the limit, substitute the value x = 1 into the simplified expression, which gives (1^2 + 2*1 + 3) / (1 - 1) = 6 / 0.
However, the expression yields an undefined result because we cannot divide by zero. This means that the limit does not exist.
3.) The limit as u approaches 1; √(1 + u) - √(1 + u^2) / (u - 1)
To simplify this expression, you can utilize the conjugate rule to rationalize the denominator. Multiply both the numerator and the denominator by the conjugate of the denominator, which is √(1 + u) + √(1 + u^2). After simplifying, the expression becomes (√(1 + u) - √(1 + u^2)) * (√(1 + u) + √(1 + u^2)) / ((u - 1) * (√(1 + u) + √(1 + u^2))).
Now, the numerator can be simplified using the difference of squares formula: a^2 - b^2 = (a - b)(a + b). Applying this formula, the numerator becomes (√(1 + u) - √(1 + u^2))(√(1 + u) + √(1 + u^2)) = (1 + u) - (1 + u^2) = u - u^2.
Simplifying further, the expression becomes (u - u^2) / (u - 1) = u(1 - u) / (u - 1).
To evaluate the limit, substitute u = 1 into the simplified expression, which yields 1(1 - 1) / (1 - 1) = 0 / 0.
Once again, we encounter an undefined result, indicating that the limit does not exist.
4.) The limit as x approaches 3 from the right; x√(x - 3)
To evaluate this limit, substitute x = 3 into the expression, which gives 3√(3 - 3) = 3√0 = 0.
Hence, the limit as x approaches 3 from the right is 0.
Remember, when evaluating limits, it's crucial to simplify the expression as much as possible and check for any restrictions. In cases where the limits yield zero in the denominator or an undefined result, the limit may not exist.