Limit related problems in differentiation [closed]
Limit related problems in differentiation [closed]
Suppose $alpha$ and $beta$ are two roots of the equation $ax^2+bx+c=0$. Find
$$lim_xtoalphafrac1-cos(ax^2+bx+c)(x-alpha)^2$$
Please help me to solve this calculus problem
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– José Carlos Santos
Sep 16 '18 at 12:01
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What kind of rules can you use?
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– Dr. Sonnhard Graubner
Sep 16 '18 at 12:07
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Instead of suggesting an edit from a different account, please use your original account and click on "edit" to add details to your question. As it stands, it is not possible if the suggested edit is by you or by someone else pretending to be you. On a side note, using your original account, you can add comments to your own posts irrespective of reputation.
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– GoodDeeds
Sep 16 '18 at 12:28
3 Answers
3
HINT
We have
$$frac1-cos(ax^2+bx+c)(x-alpha)^2=frac1-cos[a(x-alpha)(x-beta)]a^2(x-alpha)^2(x-beta)^2a^2(x-beta)^2$$
then refer to standard limit as $tto 0$
$$frac1-cos tt^2to frac12$$
Recall that $lim_t to 0 frac1-cos tt^2 = frac12$.
We have
$$frac1-cos(ax^2+bx+c)(x-alpha)^2 = frac1-cos[a(x-alpha)(x-beta)](x-alpha)^2 = frac1-cos[a(x-alpha)(x-beta)][a(x-alpha)(x-beta)]^2cdot a^2(x-beta)^2 xrightarrowxtoalpha frac12a^2(alpha-beta)^2$$
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@MathOverview Yes. Check that $$ax^2+bx+ c = aleft(x - frac-b+sqrtb^2-4ac2aright)left(x - frac-b-sqrtb^2-4ac2aright)$$
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– mechanodroid
Sep 16 '18 at 12:34
Hint. Use the Taylor series of
$$
cos(a(x-alpha)(x+beta))
$$
around $alpha$. That is
$$
cos(a(x-alpha)(x+beta))= 1-frac12(alpha-beta)^2(x-alpha)^2a^2+o((x-alpha)^3)
$$
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Welcome to Math SE! Your image is barely readable. Since the question is so short, how about type it in MathJax?
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– Toby Mak
Sep 16 '18 at 11:58