Evaluate the limit using the appropriate Limit Law(s). (If an answer does not exist, enter DNE.)
lim
t → −1
(t2 + 1)^4(t + 3)^5
((-1)^2 + 1)^4 (-1 + 3)^5
= (1+1)^4 (2)^5
= 2^4 * 2^5
= 2^9
= 512
Well, evaluating limits can be quite taxing, but let's give it a shot!
To evaluate the limit as t approaches -1 of (t^2 + 1)^4(t + 3)^5, we first note that both the numerator and denominator are raised to positive powers. This means we can directly substitute t = -1 into the given expression to find the limit.
Plugging in t = -1, we get:
((-1^2) + 1)^4((-1) + 3)^5
(0^2) * 2^5
0 * 32
0
So, the limit is equal to 0. The expression squirts out zero here, like a clown squirting water from a flower!
To evaluate the limit, we can use the Limit Law known as the Product Law. According to this law, the limit of a product is equal to the product of the limits of the individual factors, assuming the limits of the factors exist.
Let's break down the expression:
lim(t → -1) (t^2 + 1)^4 * (t + 3)^5
First, we can evaluate the limit of each factor separately:
lim(t → -1) (t^2 + 1)^4
lim(t → -1) (t + 3)^5
For the first factor, (t^2 + 1)^4, the limit can be evaluated directly by substituting the value of -1 for t:
((-1)^2 + 1)^4 = (1 + 1)^4 = 2^4 = 16
For the second factor, (t + 3)^5, the limit can also be evaluated directly by substituting -1 for t:
((-1) + 3)^5 = (2)^5 = 32
Now, we can use the Product Law to find the limit of the whole expression:
lim(t → -1) (t^2 + 1)^4 * (t + 3)^5 = lim(t → -1) (t^2 + 1)^4 * lim(t → -1) (t + 3)^5
= 16 * 32
= 512
Therefore, the limit of the expression (t^2 + 1)^4 * (t + 3)^5 as t approaches -1 is 512.
To evaluate the given limit, we can apply the Limit Law(s) to simplify the expression first.
To start, let's expand the expression using the binomial theorem. The binomial theorem states that:
(x + y)^n = (nC0)(x^n)(y^0) + (nC1)(x^(n-1))(y^1) + (nC2)(x^(n-2))(y^2) + ... + (nCn)(x^0)(y^n)
where nCk represents the binomial coefficient "n choose k".
Using the binomial theorem, we can expand (t^2 + 1)^4 as:
(t^2 + 1)^4 = (4C0)(t^2)^4(1^0) + (4C1)(t^2)^3(1^1) + (4C2)(t^2)^2(1^2) + (4C3)(t^2)^1(1^3) + (4C4)(t^2)^0(1^4)
Simplifying this expression, we get:
(t^2 + 1)^4 = (t^8) + (4t^6) + (6t^4) + (4t^2) + 1
Similarly, we can expand (t + 3)^5 as:
(t + 3)^5 = (5C0)(t^5)(3^0) + (5C1)(t^4)(3^1) + (5C2)(t^3)(3^2) + (5C3)(t^2)(3^3) + (5C4)(t^1)(3^4) + (5C5)(t^0)(3^5)
Simplifying this expression, we get:
(t + 3)^5 = (t^5) + (15t^4) + (90t^3) + (270t^2) + (405t) + 243
Now, let's substitute these expanded expressions back into the original limit:
lim t → -1 (t^2 + 1)^4 * (t + 3)^5
= lim t → -1 ((t^8) + (4t^6) + (6t^4) + (4t^2) + 1) * ((t^5) + (15t^4) + (90t^3) + (270t^2) + (405t) + 243)
Next, we can evaluate the limit by plugging in the value t = -1 into the expression:
((-1)^8) + (4(-1)^6) + (6(-1)^4) + (4(-1)^2) + 1) * ((-1)^5 + (15(-1)^4) + (90(-1)^3) + (270(-1)^2) + (405(-1)) + 243)
Simplifying this expression, we obtain:
(1) + (4) + (6) + (4) + 1) * ((-1) + (15) + (-90) + (270) + (-405) + 243)
= 16 * (-1 + 15 - 90 + 270 - 405 + 243)
= 16 * (-3)
Therefore, the value of the given limit is -48.