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GrogVix [38]
3 years ago
13

Guess the value of the limit (correct to six decimal places). (if an answer does not exist, enter dne.) lim hâ0 (4 + h)5 â 1024

h
Mathematics
1 answer:
kupik [55]3 years ago
4 0
\displaystyle\lim_{h\to0}\frac{(4+h)^5-1024}h=\lim_{h\to0}\frac{(4+h)^5-4^5}h

Recall the definition of the derivative of a function f(x) at a point x=c:

f'(c)=\displaystyle\lim_{h\to 0}\frac{f(c+h)-f(c)}h

We can then see that f(c)=c^5, and by the power rule we have f'(c)=5c^4. Then replacing c=4, we arrive at

\displaystyle\lim_{h\to0}\frac{(4+h)^5-4^5}h=5\times4^4=1280

Alternatively, we could have expanded the binomial, giving

\dfrac{(4+h)^5-4^5}h=\dfrac{(4^5+5\times4^4h+10\times4^3h^2+10\times4^2h^3+5\times4h^4+h^5)-4^5}h
=\dfrac{1280h+640h^2+160h^3+20h^4+h^5}h
=1280+640h+160h^2+20h^3+h^4

and so as h\to0 we're left with 1280, as expected.
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