Estimate the following limit by substituting smaller and smaller values of h.
lim 8^h-1/h
h->0
Why did the mathematician bring a ladder to the equation?
Because they were going to "lim(h -> 0)"!
But in all seriousness, let's dive into finding the limit of 8^h-1/h as h approaches 0 by substituting smaller and smaller values of h.
For the numerator, 8^h - 1, as h gets closer and closer to 0, 8^h will tend towards 1 since any number raised to the power of 0 is equal to 1. So the numerator, 8^h - 1, will get very close to 1 - 1 = 0.
For the denominator, h, as h gets smaller and smaller, the value of h approaches 0.
Now, when we divide a small number (the numerator approaching 0) by an even smaller number (the denominator approaching 0), the result will approach infinity. Therefore, we can conclude that:
lim(8^h-1/h) as h->0 = ∞
Just don't divide by 0 in any real calculations, or you might cause a mathematical meltdown!
To estimate the limit as h approaches 0 for the expression (8^h - 1)/h, we can substitute smaller and smaller values of h and observe the pattern.
Let's start by substituting a small value for h, such as 0.1:
(8^0.1 - 1)/0.1 = (1.39794 - 1)/0.1 = 0.39794
Now, let's try substituting an even smaller value for h, such as 0.01:
(8^0.01 - 1)/0.01 = (1.08329 - 1)/0.01 = 0.08329
As you can see, as we substitute smaller values of h, the expression seems to converge to a value. However, to get a more accurate estimate, let's continue with a few more calculations:
For h = 0.001: (8^0.001 - 1)/0.001 = 0.008313
For h = 0.0001: (8^0.0001 - 1)/0.0001 = 0.00083296
For h = 0.00001: (8^0.00001 - 1)/0.00001 = 0.000083298
Based on our calculations, it appears that the expression (8^h - 1)/h approaches approximately 0.000083298 as h approaches 0. Hence, we can estimate the limit as h approaches 0 to be approximately 0.000083298.
To estimate the limit as h approaches 0 for the function f(h) = (8^h - 1)/h, we can substitute smaller and smaller values of h and observe the trend in the resulting values.
Let's start by substituting some small values of h and calculating the corresponding values for f(h):
For h = 0.1, f(h) = (8^0.1 - 1) / 0.1 ≈ 0.378
For h = 0.01, f(h) = (8^0.01 - 1) / 0.01 ≈ 0.368
For h = 0.001, f(h) = (8^0.001 - 1) / 0.001 ≈ 0.367
As we substitute smaller values of h, we can observe that the resulting values are approaching a certain number, which suggests that there might be a limit as h approaches 0.
To get a more accurate estimate, let's continue substituting even smaller values of h:
For h = 0.0001, f(h) = (8^0.0001 - 1) / 0.0001 ≈ 0.3679
For h = 0.00001, f(h) = (8^0.00001 - 1) / 0.00001 ≈ 0.3678
The trend we observe is that as we substitute smaller and smaller values of h, the values of f(h) are approaching approximately 0.3678. This suggests that the limit as h approaches 0 for the given function is approximately 0.3678.
However, please note that this method of estimating the limit by substituting smaller and smaller values of h is not a rigorous proof. To formally prove the limit, one needs to use the definition of a limit or apply appropriate mathematical techniques like L'Hôpital's rule or Taylor series expansions.
If you really mean what you typed
8^h - (1/h)
h 8^h / h - 1/h
[h 8^h - 1 ]/h
8^0 = 1 so
[0 - 1] /0 --- > oo