◇解説◇ ○高校の微分積分で漸近線の問題が登場するのは,微分法の応用として,「増減,極値,凹凸,変曲点,漸近線の方程式を求めてグラフの概形を書け」という場面です。 したがって,漸近線の方程式を単独で問うことはまれです。 ○漸近線とは,一言で言えば「ある曲線が限りなく近づく相手方の線」のことですが,高校数学で漸近線と言えば「直線」に限ります(*)。したがって,
高校数学の漸近線は
(A)縦線:「x軸に垂直な直線 ( x=a の形のもの)」
の2種類だけです。(B)斜め線:「y=ax+bの形のもの(a=0の場合:横線を含む)」 (*) 高校数学では,直線以外の漸近線は考えません。 例えば曲線y = x2+ ![]()
有限の値 a に対して,
limx遶奇ソス+af(x) = ∞ または −∞ limx遶奇ソス−af(x) = ∞ または −∞ となるとき(一方だけでもよい), x = a が漸近線です. この形の漸近線は,「分母が0となるxの値」を探せばほとんど見つかります。( y = tan x における x = ![]() 例
y=
![]() 分母が0となるxの値は 2, -4 です。 (x = 1 , x = -3 は漸近線ではなく,x軸との交点です.) |
(B) ≪「斜め又は横」の漸近線の求め方≫ x → +∞ のとき y = ax+b が漸近線であるとは, limx遶奇ソス+∞{f(x) -(ax+b)} = 0・・・(1) となることです。この条件を満たすa, b の値は,2段階に分けて求められます. (1)が成り立つならば当然 limx遶奇ソス+∞ { ![]() ![]() したがって, limx遶奇ソス+∞ ![]() その求めた a の値を用いて, limx遶奇ソス+∞ {f(x)−ax} = b ・・・(3) x → -∞ のときも同様にして求めます.
1) limx遶奇ソス+∞
![]() この極限がなければ,x → ∞ のとき漸近線なし。 2) 1)の極限値が存在するとき,a の値を用いて, limx遶奇ソス+∞ {f(x)−ax} = b とする.
※ 横(x軸に平行)の漸近線は,直ちに見つかることがあります。
例 y = e−x2のとき limx遶奇ソス+∞e−x2 = 0だから y = 0 は漸近線です. 上の議論で a = 0, b= 0 です。 ※ (y = ±f(x) のように初めから曲線が2つあるような場合は論外として) 横の漸近線があれば斜めの漸近線はありません. だから,右側(→∞)について,横の漸近線が見つかれば,斜めの漸近線を調べる必要はありません. 左側(→−∞)については,右側とは別の漸近線が存在することもありますが,左側で横または斜めの漸近線が1つ見つかれば,左側で他の漸近線を調べる必要はありません. |
(例1)
y = ![]() ◇x軸に垂直な漸近線◇ x = 0, 1 のとき分母が 0 となり,分子は0とならない. limx遶奇ソス0 ![]() ![]() だから,漸近線の方程式はx = 0, 1・・・ 答 ◇x→±∞のときの漸近線◇ limx遶奇ソス±∞ ![]() 漸近線の方程式は y = 1・・・答
(例2)
y = ![]() ◇x軸に垂直な漸近線◇ x = ± 1 のとき分母が 0 となり,分子は0とならない. limx遶奇ソス-1 ![]() ![]() 漸近線の方程式は x = ± 1 ・・・ 答 ◇x→±∞のときの漸近線◇ limx遶奇ソス±∞ ![]() limx遶奇ソス±∞ { ![]() ![]() ![]() だから,漸近線の方程式は y = 2x ・・・ 答
(例3)
y = e ![]() ![]() x = 0 のとき指数の分母が 0 となり, limx遶奇ソス+0 e ![]() limx遶奇ソス−0 e ![]() アより漸近線の方程式は x = 0 ・・・ 答 (イは漸近線とはならない) ◇x→±∞のときの漸近線◇ limx遶奇ソス±∞ e ![]() だから,漸近線の方程式は y = 1 ・・・ 答 |
■ 問題 次の曲線の漸近線の方程式を求めなさい。
○空欄にはスペースを使わずに半角の「アルファベット小文字または数字」だけを使用するものとします.
○HELPが必要なときは, ![]()
(1)
y = x+ ![]()
(2)
y = ![]()
(3)
y = ![]()
(4) y = ![]() |
[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][漸近線の方程式について/16.12.24]
わかりやすい説明で概要を掴めますがいざ実践してみるとなかなかできないものです。
なので簡単な問題がその後に付属して実戦形式で行えるのは良かったと思います。
それによって理解が深まった気がします。
=>[作者]:連絡ありがとう.筆者の苦労が込められているという点からいえば増減.極値.凹凸.変曲点.漸近線.グラフの方もお勧めしたいところです. |
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