Limit question

Question

Limit question

abbles · Answer

Each limit represents the derivative of some function f at some number a.  State such an f and a in each case.
$$\frac{ (1+h)^10 - 1 }{ h }$$ As h approaches 0.

abbles · Answer

That should be to the 10th power

marcelie · Answer

i think u can plug in 0  for h

abbles · Answer

It would give 0 for the denominator, which is undefined...

marcelie · Answer

hmmm

abbles · Answer

It says the answer is $$f(x) = x^10, a = 1 $$ or $$f(x) = (1+x)^10, a = 0$$
Can anyone explain this to me?

agent0smith · Answer

$$\large f'(a+h) =\lim_{h \rightarrow 0} \frac{  f(a+h) - f(a) }{ h }$$ $$\large f(a+h) = \lim_{h \rightarrow 0}\frac{ (1+h)^{10} - 1 }{ h }$$You just kinda have to compare.

agent0smith · Answer

should be f'(a+h) there

a has to be 1, right? Just by comparing the two

f(x) looks like it has to be x^10 cos then $$\large f(a+h) = (a+h)^{10}$$

marcelie · Answer

good job  x'd @agent0smith

agent0smith · Answer

ty sleek-feathered one

agent0smith · Answer

Make sense @Abbles? you can pretty much ignore the  f(a) part which is equal to 1. 

Just compare $\large f(a+h)$ in the top one, to $\large (1+h)^{10} $ in the bottom one

abbles · Answer

I still really don't get this one...

phi · Answer

As agent said, this is a "pattern matching problem"
We start with the definition of how to find the derivative of f(x):
$$ f'(x) = \lim_{h\rightarrow 0}\frac{  f(x+h) - f(x) }{ h }$$
at some particular value x=a, we get the expression
$$ f'(a) = \lim_{h\rightarrow 0}\frac{  f(a+h) - f(a) }{ h }$$

Now the question
***
Each limit represents the derivative of some function f at some number "a"
***
$$ f'(1) = \lim_{h\rightarrow 0}\frac{  (1+h)^{10} - 1 }{ h }$$
we have to make that expression match up with the definition of the derivative.
notice the bottom is "h", so that looks good.
the top has   (1+h)^10  and that should match up with f(a+h)
that suggests that f(a+h)= (1+h)^10, with a=1
which would mean f(a) = a^10 = 1^10 

in other words, if 
$$ f(x)= x^{10} $$
we would find its derivative as
$$ f'(x) = \lim_{h\rightarrow 0}\frac{  (x+h)^{10} - x^{10} }{ h }$$
and if we want the derivative at x=1, that becomes
$$ f'(1) = \lim_{h\rightarrow 0}\frac{  (1+h)^{10} - 1^{10} }{ h } \
= \lim_{h\rightarrow 0}\frac{  (1+h)^{10} - 1 }{ h }
$$
which matches your limit problem