Estimating a limit with smaller and smaller quantities.

Question

Estimate the following limit by substituting smaller and smaller values of h

lim h->0 ((7+h)^3 - 343)/h

(Your answer should be accurate within 0.001.)

I don't understand what I'm supposed to do with this.

This?

(7+.001)^3 -343/.001

That gave me the wrong answer

Answer 1

lim h->0 ((7+h)^3 - 343)/h

This is mathematical notation which means that the expression ((7+h)^3 - 343)/h approaches a number, called the limit, getting closer and closer, but never reaching the limit as h approaches 0. Note that we must estimate this value because the expression gets closer and closer to it as h approaches 0 and the expression is undefined for h=0.

Now, we must realize that estimates are not exact, but are precise enough to use. For example, 3.14 is a close enough estimate for ∏ to solve many problems. If you need a better estimate, use 3.14159. And, if that is not exact enough, add more and more digits. SPECIAL NOTE: 3,14 is a value for ∏ that is accurate within 0.01. The problem does not ask you to find the value of h accurate with 0.001, but to find the value of the expression ((7+h)^3 - 343)/h within 0.001. Please don't solve the wrong problem !!

"(Your answer should be accurate within 0.001.)" means that we must make h close enough to 0 that the value of the expression no longer changes with more digits. For example, 147.000xxxx no longer changes with more digits.

h                  ((7+h)^3 - 343)/h      change

1                  169.000000000           

0.5               157.750000000           11.250000

0.1               149.110000000           8.640000

0.05             148.052500000           1.057500

0.01             147.210100000           0.842400

0.005           147.105025000           0.105075

0.001           147.021001000           0.084024

0.0005         147.010500250           0.010501

0.0001         147.002100010           0.008400

0.00005       147.001050002           0.001050

0.00001       147.000209995           0.000840

0.000005     147.000104994           0.000105

0.000001     147.000021002           0.000084

Our estimate of the limit is 147.000. This is because the change (or difference from the previous value) is 0,000840 (which is less than the required accuracy of 0.001). With additional digits, a more exact estimate would look like 147.000xxxxxxx.

ALSO NOTE that a value of 147 is only accurate within 1. An estimate of 147.0 is only accurate within 0.1. An estimate of 147,00 is only accurate within 0.01. An estimate of 147.000 is only accurate within 0,001.

Answer 2

You have to consider function f(x) = x³. Then if x₀ = 7 x₁ = 7 + h. So f(7 + h) - f(7) ≈ f'(7)·h = 3·7²h = 147h,

so [(7 + 0.001)³ - 7³]/0.001 ≈ 147 (take h = 0.001)

Answer 3

You need to try smaller values of h, then. The value at .001 gives you a good idea of what the limit is supposed to be of course, as you get 147.021. You might guess that the limit will be 147, so you need to have smaller values of h. At h = .0001, you get 147.0021. Note how the hypothesized difference from 147 went from .021 to .0021 as h went from .001 to .0001; the difference appears to be proportional to h. We could either try to get very close to the .001 error or just use h = .00001. This gives us 147.00021, within .001 of 147, as the difference is .00021l, so h = .00001 is small enough.

If you expand ((7+h)^3-7^3) /h, you get (343 + 147 h + 21 h^2 + h^3 - 343)/h = 147 + 21h + h^2. This has limit 147 as h goes to 0. Also notice the linear term of 21h, which is what our error approximates (if we showed more decimal places, we would see that it is not exactly this). Thus, if we solve 21h = .001, we would need to let h be just slightly smaller to achieve the error. If you feel better about being closer to an error of .001, you could pick .00004, instead.

Answer 4

If you plug in smaller and smaller numbers close to zero (like .001, .0001, .00001) in for h, you should see that the limit is getting closer and closer to 147. To see this in the equation look at it this way- Since you can’t plug in zero for x (that would put a zero in the denominator and you can’t divide by zero) so to estimate the limit, expand the function first, simplify, then plug in zero.

(7+h)^3-343/h = 343+98h+7h^2+49h+14h^2+h^3-343/h

combine like terms (the 343 will cancel) and factor out an h in the numerator to get

h(147+28h+h^2)/h

the h in the numerator and denominator will cancel out giving you

147+28h+h^2

with the exclusion the h does not equal zero

then plug in zero for h in 147+28h+h^2

147+28(0)+(0)^2 which equals 147. So 147 is the estimation of the limit as h approaches 0