How do I evaluate lim as h->0 [5((1/2)+h)^4 - 5(1/2)^4]/h
by definition
dy/dx = lim (f(x+h) - f(x))/h as h ---> 0
if you look at your expression carefully, you can see that they are attempting to find dy/dx of f(x) = 5x^4 when x=1/2
so dy/dx = 20x^3, which at x= 1/2 = 20/8 = 5/2
just like in Damon's answer.
Also , another similar question that I am having trouble with is lim as x->0 (tan^3 (2x))/(x^3)
binomial expansion of (.5+h)^4
1* .5^4 + 4* .5^3 h + 6* .5^2 h^2 etc
5 [ .5^4 + 4* .5^3 h + 6* .5^2 h^2 + 4 .5 h^3 + h^4 -.5^4 ] / h
only term left when h--> 0 is
5 [ 4*.5^3 ]
I figured maybe the derivative is in the next chapter :)
Thanks for the help, I know what to do now. :D
To evaluate the limit as h approaches 0 for the given expression, we'll need to simplify and apply some algebraic manipulations. Here's how you can do it step by step:
Step 1: Expand the numerator
Start by expanding the first term in the numerator using the binomial theorem. The binomial theorem states that (a + b)^n can be expanded as:
(a + b)^n = C(n, 0)a^n * b^0 + C(n, 1)a^(n-1) * b^1 + C(n, 2)a^(n-2) * b^2 + ... + C(n, n-1)a^1 * b^(n-1) + C(n, n)a^0 * b^n
where C(n, k) is the binomial coefficient and is given by C(n, k) = n!/[k!(n-k)!].
In our case, we have (1/2 + h)^4. Expanding it using the binomial theorem, we get:
(1/2 + h)^4 = C(4, 0)(1/2)^4 * h^0 + C(4, 1)(1/2)^3 * h^1 + C(4, 2)(1/2)^2 * h^2 + C(4, 3)(1/2)^1 * h^3 + C(4, 4)(1/2)^0 * h^4
This simplifies to:
(1/2 + h)^4 = (1/2)^4 + 4(1/2)^3 * h + 6(1/2)^2 * h^2 + 4(1/2)^1 * h^3 + (1/2)^0 * h^4
So, the numerator becomes:
5((1/2 + h)^4) = 5((1/2)^4 + 4(1/2)^3 * h + 6(1/2)^2 * h^2 + 4(1/2)^1 * h^3 + (1/2)^0 * h^4)
= 5((1/16) + (4/8) * h + (6/4) * h^2 + (4/2) * h^3 + h^4)
= 5((1/16) + (1/2) * h + (3/2) * h^2 + 2 * h^3 + h^4)
Step 2: Simplify the rest of the expression
Now, simplify the rest of the expression by expanding the second term in the numerator and dividing it by h:
5(1/2)^4 = 5(1/16) = 5/16
So, the whole numerator becomes:
5((1/2 + h)^4) - 5(1/2)^4 = 5((1/16) + (1/2) * h + (3/2) * h^2 + 2 * h^3 + h^4) - 5/16
Step 3: Divide the numerator by h
Next, divide the numerator expression by h:
[5((1/16) + (1/2) * h + (3/2) * h^2 + 2 * h^3 + h^4) - 5/16] / h
Step 4: Simplify the expression further
Distribute the factor of 5 to each term in the numerator:
[5(1/16) + 5(1/2) * h + 5(3/2) * h^2 + 5(2 * h^3) + 5(h^4) - 5/16] / h
Simplify each term:
(5/16 + (5/2) * h + (15/2) * h^2 + 10 * h^3 + 5 * h^4 - 5/16) / h
Step 5: Combine like terms
Combine the like terms in the numerator:
[(5/16 - 5/16) + (5/2) * h + (15/2) * h^2 + 10 * h^3 + 5 * h^4] / h
Simplify the numerator:
[(5/2) * h + (15/2) * h^2 + 10 * h^3 + 5 * h^4] / h
Step 6: Cancel out h
Now, cancel out h from the numerator and denominator:
(5/2) + (15/2) * h + 10 * h^2 + 5 * h^3
Step 7: Find the limit as h approaches 0
Finally, substitute h = 0 into the expression:
lim as h->0 [(5/2) + (15/2) * h + 10 * h^2 + 5 * h^3]
Since there are no h terms left in the expression, all the terms containing h approach 0 as h approaches 0. Therefore, the limit becomes:
lim as h->0 [(5/2)]
Hence, the value of the limit as h approaches 0 for the given expression is 5/2.