![]() ![]() *** 科目 *** 数Ⅰ・A数Ⅱ・B数Ⅲ高卒・大学初年度 *** 単元 *** 複素数平面二次曲線媒介変数表示と極座標 数列の極限関数導関数不定積分定積分 行列1次変換 ※高校数学Ⅲの「関数・グラフ」について,このサイトには次の教材があります.
この頁へGoogleやYAHOO ! などの検索から直接来てしまったので「前提となっている内容が分からない」という場合や「この頁は分かったがもっと応用問題を見たい」という場合は,他の頁を見てください. が現在地です. |
∞−∞,
のように相反する向きに引っ張り合っているような場合をいい,結果が不定になるということではありません。![]() ![]() →→ 不定形の極限では,式を変形して強弱が分かる形に直してから極限を求めます。 ○ これに対して,式が見かけ上,
∞+∞,∞×∞,
のように大きくなるものばかりのときや,
![]()
0×0,
のように小さくなるものばかりの組合せのときは不定形とはいいません。![]() →→ そのままの形で直ちに極限が求まります。 ■ 不定形の例 limx→1■ 不定形ではないもの limx→∞ (x2+x) は ∞+∞形だから, |
■即答問題 次のうち,いわゆる「不定形の極限」となっているものを選びなさい。 (複数回答:該当するものすべてにチェックをつける) ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
【解説】 limx→∞ ![]() ア) 分母の次数 > 分子の次数 ならば 0 イ) 分母の次数 < 分子の次数 ならば (符号に応じて)+∞または-∞ ウ) 分母の次数 = 分子の次数 ならば 最高次の項の係数の比 となります. (考え方) ア) limx→∞ ![]() のように分母の次数が高いときは,約分すると分母だけに x が残り, 分母→∞で,結果は0となります. limx→∞ ![]() ![]() イ) limx→∞ ![]() のように分子の次数が高いときは,約分すると分子だけに x が残り ます.この場合,係数の符号に応じて±∞のいずれかになります. limx→∞ ![]() limx→∞ ![]() ウ) limx→∞ ![]() のように分母分子の次数が等しいときは,約分で係数だけが残り ますので,係数の比になります. limx→∞ ![]() ![]() limx→∞ ![]() のように,分母分子が多項式になっているときは,「各々の最大項でくくる」と,左の考え方に帰着できます. 例 ア) limx→∞ ![]() =limx→∞ ![]() ![]() ![]() ![]() ![]() ![]() ![]() イ) limx→∞ ![]() =limx→∞ ![]() ![]() ![]() ![]() ![]() ![]() ![]() ウ) limx→∞ ![]() =limx→∞ ![]() ![]() ![]() ![]() ![]() ![]() ![]() ※ 上の式で,( )内は1になります. 以上により,(参考)の初めに青字でまとめたア)~ウ)がいえます.これらを用いて以下のような問題を解くことができます. |
■問題1 次の空欄を埋めなさい。 採点するやり直す解説 採点するやり直す解説 分母が0になる原因となっている式x−1を約分によって取り除いてから,x=1を代入します. 採点するやり直す解説 分母が0になる原因となっている式x−2を約分によって取り除いてから,x=2を代入します. 採点するやり直す解説 分母と分子をそれぞれの最大次数でくくるとよいでしょう. 採点するやり直す解説 分母と分子をそれぞれの最大次数でくくるとよいでしょう. 採点するやり直す解説 分母と分子をそれぞれの最大次数でくくるとよいでしょう. |
例 1 limx→∞ ( ![]() = limx→∞ ![]() ![]() = limx→∞ ![]() ![]() = limx→∞ ![]() ![]() ![]() = ![]() 【要点】 無理関数の極限 → 「分子」の有理化も考える |
次の空欄を埋めなさい。 |
例 2 limx→-∞ ( ![]() = lims→∞ ( ![]() = lims→∞ ![]() ![]() = lims→∞ ![]() ![]() = 0 (∵分子→2,分母→∞) 【要点】 x→−∞ のとき「x =−s とおくと s → ∞」 |
問 2 |
例 3 limx→1 ![]() ![]() = limx→1 ![]() ![]() = limx→1 ( ![]() = 2 【要点】 分母が見かけ上0となるとき → 「約分によって,0となる原因を取り除く」 |
問 3 |
[注]直前にPC版から入られた場合は,自動転送でスマホ版に来ていますので,ブラウザの[戻るキー]では戻れません(堂々巡りになる).下記のリンクを使ってメニューに戻ってください.
![]() ![]() |
■[個別の頁からの質問に対する回答][関数の極限1について/18.9.26]
大学卒業にあたり,数学3の極限分野の知識が必要になった.そこで使わせていただいたが,非常に要点を得てまとめられていたので,復習に役立った.統一の取れたサイト構成でしたので,今後も利用していきたい.
■[個別の頁からの質問に対する回答][関数の極限について/17.5.9]
=>[作者]:連絡ありがとう. 不定形の極限のところが分かりやすくとても良いと思う
■[個別の頁からの質問に対する回答][関数の極限1について/17.3.7]
=>[作者]:連絡ありがとう. 三角関数の極限を載せてほしい
■[個別の頁からの質問に対する回答][関数の極限1について/17.2.9]
=>[作者]:連絡ありがとう.三角関数の極限の頁を見てください.こちら 復習になりました
ありがとうございます☺
■[個別の頁からの質問に対する回答][関数の極限について/17.1.21]
=>[作者]:連絡ありがとう. いつも分かりやすい説明で、よく利用させてもらっています。
このページで、いくつかの問題を解く際に、答え欄をクリックしても反応しないために解答できなかった問題が3つほどありました。私はMac OSを使っていて、Safariを重荷利用していますが、たまに他のサイトを利用する際にSafariでは動かなくても、Google chlome では動く場合があり、こちらのページもGoogle Chlomeでも試してみましたが、やはり動きませんでした。(厳密に言うと、即答問題の上列左から2番目と4番目をクリックできないことと、問題1の(6)に答えを入力できないことです。)あまり支障をきたすわけではないのですが、気になったのでコメントしました^^
でも、基本的にどのトピックも分かりやすい説明で、とても助かっています。ありがとうございます。
=>[作者]:連絡ありがとう.ブラウザによってはチェックボックスや入力欄が左の関数の陰に入ってしまって下敷きになっていたようですので訂正しました. |
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