Are these correct?
lim x->0 (x)/(sqrt(x^2+4) - 2)
I get 4/0= +/- infinity so lim x->0+ = + infinity? and lim x->0- = + infinity?
lim x->1 (x^2 - 5x + 6)/(x^2 - 3x + 2)
I get 2/0, so lim x-> 1+ = - infinity? and lim x->1- = + infinity?
lim h->0 [(-7)/(2+h^2) + (7/4)]/h
I used a computational website to get (7/4) as the answer, but I did not get this. My work ends with: (-28)/(4(h+8) + (7(h+8)/(4(h+8)) and I end with -5.25 for an answer?
Last one!
lim x-> neg. infinity (-2x^2 + 3x - 2)/(5x^3 + 4x -x + 1)
Don't even know about this one?
Please help. I would like to understand these.
Sure, let's go through each of these problems step by step to figure out the correct solutions.
1. lim(x->0) (x)/(sqrt(x^2+4) - 2)
To find this limit, we can try to simplify the expression. We start by multiplying the numerator and denominator by the conjugate of the denominator, which is (sqrt(x^2+4) + 2):
lim(x->0) [x * (sqrt(x^2+4) + 2)] / [sqrt(x^2+4) - 2] * [sqrt(x^2+4) + 2]
Next, we simplify the numerator:
lim(x->0) [x * (sqrt(x^2+4) + 2)] = x * [sqrt(x^2+4) + 2] = xsqrt(x^2+4) + 2x
Now, we can cancel out the common factor of x in both the numerator and denominator:
lim(x->0) [sqrt(x^2+4) + 2]
As x approaches 0, the function inside the square root becomes sqrt(4) = 2. Therefore, the limit is:
lim(x->0) [sqrt(x^2+4) + 2] = sqrt(2) + 2
So the correct answer is sqrt(2) + 2.
2. lim(x->1) (x^2 - 5x + 6)/(x^2 - 3x + 2)
Again, let's try to simplify the expression first. This time, we can factor both the numerator and denominator:
(x^2 - 5x + 6)/(x^2 - 3x + 2) = [(x - 2)(x - 3)]/[(x - 1)(x - 2)]
Now, we can cancel out the common factor of (x - 2) in both the numerator and denominator:
[(x - 2)(x - 3)]/[(x - 1)(x - 2)] = (x - 3)/(x - 1)
As x approaches 1, we get:
lim(x->1) (x - 3)/(x - 1) = (1 - 3)/(1 - 1) = -2/0
Here, we can see that we have an indeterminate form of -2/0, which means we need to further simplify the expression to find the limit. Taking a closer look, as x approaches 1 from the left side (x -> 1-), the denominator (x - 1) becomes negative while the numerator (x - 3) remains positive. So the limit from the left side is -∞.
On the other hand, as x approaches 1 from the right side (x -> 1+), the denominator (x - 1) becomes positive while the numerator (x - 3) remains negative. So the limit from the right side is +∞.
Therefore, the correct answer is lim(x->1-) = -∞ and lim(x->1+) = +∞.
3. lim(h->0) [(-7)/(2+h^2) + (7/4)]/h
To evaluate this limit, we need to first simplify the expression. Let's start by combining the fractions inside the square brackets:
[-7/(2+h^2) + (7/4)] = [-7(4)/((2+h^2)(4)) + (7(2+h^2))/((4)(2+h^2))]
= (-28/(4(2+h^2))) + (7(2+h^2))/(4(2+h^2))
Now, let's combine the fractions over a common denominator:
= (-28 + 7(2+h^2)) / (4(2+h^2))
= (-28 + 14 + 7h^2) / (4(2+h^2))
= (14 + 7h^2) / (4(2+h^2))
Now, let's divide this whole fraction by h:
= [(14 + 7h^2) / (4(2+h^2))] / h
= (14 + 7h^2)/(h(4(2+h^2)))
= (7(2 + h^2))/(h(4(2+h^2)))
Notice that we can cancel out the common factor of (2 + h^2) in both the numerator and denominator:
= 7/h
As h approaches 0, we get:
lim(h->0) 7/h = 7/0
This is an indeterminate form of 7/0, which means we need to further evaluate. In this case, as h approaches 0, the function 7/h becomes infinitely large. Therefore, the limit is +∞.
So the correct answer is +∞.
4. lim(x->-∞) (-2x^2 + 3x - 2)/(5x^3 + 4x -x + 1)
To evaluate this limit, we can divide both the numerator and denominator by x^3:
= [(-2x^2/x^3) + (3x/x^3) - (2/x^3)] / [(5x^3/x^3) + (4x/x^3) - (x/x^3) + 1/x^3]
= [(-2/x) + (3/x^2) - (2/x^3)] / [5 + (4/x^2) - (1/x^3) + (1/x^3)]
As x approaches -∞, all the terms with x in the denominator become 0, and the expression simplifies to:
= [-2/-∞ + 3/(-∞)^2 - 2/(-∞)^3] / [5 + 4/(-∞)^2 - 1/(-∞)^3 + 1/(-∞)^3]
= [0 + 0 - 0] / [5 + 0 - 0 + 0] = 0/5 = 0
So the correct answer is 0 when x approaches -∞.