【高校数学Ⅲ】2-2-2 いろいろな関数の極限｜問題集

Q: 三角関数の極限で重要な公式は何ですか？

最も重要なのは lim_x→0 (sinx / x) = 1 です。これを基にして、lim_x→0 ((1 - cosx) / x^2) = 1/2 や、角度の置換による応用問題が多数出題されます。

Q: 指数・対数・三角関数を組み合わせた極限はどう解くの？

それぞれの関数の性質をうまく組み合わせて解きます。たとえば lim_x→0 ((1 + x)^{1/x}) = e や lim_x→0 ((sinx)/x)^{1/x^2} などでは、指数関数・対数関数の変形を組み合わせて計算します。

Q: 難しい極限問題を解くときのコツは？

①式の形を見てどの関数の極限かを判断する、②0/0や∞/∞などの不定形を見つけて変形を試みる、③指数・対数・三角関数の代表的な極限公式に当てはめる、という3ステップで整理して考えるのが効果的です。

Question 1

(1)\(\displaystyle \lim_{x\to-\infty}\left(\frac{1}{2}\right)^x\)

Answer

\(=\infty\)

Question 2

(2)\(\displaystyle \lim_{x\to\infty}\left(\frac{1}{3}\right)^x\)

Question 3

(3)\(\displaystyle \lim_{x\to\infty}2^{-3x}\)

Question 4

(4)\(\displaystyle \lim_{x\to-\infty}2^{-2x}\)

Answer

\(=\infty\)

Question 5

(5)\(\displaystyle \lim_{x\to\infty}\frac{3^x+4^x}{5^x-2}\)

Answer

\(\displaystyle =\lim_{x\to\infty}\frac{(\frac{3}{5})^x+(\frac{4}{5})^x}{1-2・(\frac{1}{5})^x}\)
\(=0\)

Question 6

(6)\(\displaystyle \lim_{x\to+0}\log_{\frac{1}{3}}\frac{1}{x}\)

Answer

\(=-\infty\)

Question 7

(7)\(\displaystyle \lim_{x\to+0}\log_{0.5}x\)

Answer

\(=\infty\)

Question 8

(8)\(\displaystyle \lim_{x\to\infty}\log_{2}\frac{4x-1}{x+2}\)

Answer

\(\displaystyle =\lim_{x\to\infty}\log_{2}\frac{4-\frac{1}{x}}{1+\frac{2}{x}}\)
\(=2\)

Question 9

(9)\(\displaystyle \lim_{x\to\infty}\cos{\frac{1}{x}}\)

Question 10

(10)\(\displaystyle \lim_{x\to-\frac{3\pi}{2}+0}\tan x\)

Answer

\(=-\infty\)

Question 11

(11)\(\displaystyle \lim_{x\to\pi}\tan x\)

Question 12

(12)\(\displaystyle \lim_{x\to\infty}\frac{\cos x}{x}\)

Answer

\(-1\leqq\cos x\leqq1\)より、
\(\displaystyle -\frac{1}{x}\leqq\frac{\cos x}{x}\leqq\frac{1}{x}\)
\(\displaystyle \lim_{x\to\infty}-\frac{1}{x}\leqq\lim_{x\to\infty}\frac{\cos x}{x}\leqq\lim_{x\to\infty}\frac{1}{x}\)
\(\displaystyle 0\leqq\lim_{x\to\infty}\frac{\cos x}{x}\leqq0\)
よって、
\(\displaystyle \lim_{x\to\infty}\frac{\cos x}{x}=0\)

Question 13

(13)\(\displaystyle \lim_{x\to0}x\sin\frac{1}{x}\)

Answer

\(\displaystyle 0\leqq\left|\sin\frac{1}{x}\right|\leqq1\)より、
\(\displaystyle 0\leqq|x|\left|\sin\frac{1}{x}\right|\leqq|x|\)
\(\displaystyle 0\leqq\lim_{x\to0}\left|x\sin\frac{1}{x}\right|\leqq\lim_{x\to0}|x|\)
\(\displaystyle 0\leqq\lim_{x\to0}x\sin\frac{1}{x}\leqq0\)
よって、
\(\displaystyle \lim_{x\to0}x\sin\frac{1}{x}=0\)

Question 14

(14)\(\displaystyle \lim_{x\to0}\frac{\sin3x}{x}\)

Answer

\(\displaystyle =\lim_{x\to0}3・\frac{\sin3x}{3x}\)
\(=3\)

Question 15

(15)\(\displaystyle \lim_{x\to0}\frac{\sin2x}{3x}\)

Answer

\(\displaystyle =\lim_{x\to0}\frac{\sin2x}{2x}・\frac{2x}{3x}\)
\(\displaystyle =\frac{2}{3}\)

Question 16

(16)\(\displaystyle \lim_{x\to0}\frac{\tan x}{x}\)

Answer

\(\displaystyle =\lim_{x\to0}\frac{\sin x}{x}・\frac{1}{\cos x}\)
\(=1\)

Question 17

(17)\(\displaystyle \lim_{x\to0}\frac{1-\cos2x}{x^2}\)

Answer

\(\displaystyle =\lim_{x\to0}\frac{2\sin^2x}{x^2}\)
\(\displaystyle =\lim_{x\to0}2\left(\frac{\sin x}{x}\right)^2\)
\(=2\)

Question 18

(18)\(\displaystyle \lim_{x\to0}\frac{\sin3x}{\sin5x}\)

Answer

\(\displaystyle =\lim_{x\to0}\frac{\sin3x}{3x}・\frac{5x}{\sin5x}・\frac{3x}{5x}\)
\(\displaystyle =\frac{3}{5}\)

Question 19

(19)\(\displaystyle \lim_{x\to0}\frac{1-\cos x}{x^2}\)

Answer

\(\displaystyle =\lim_{x\to0}\frac{1-\cos^2x}{x^2(1+\cos x)}\)
\(\displaystyle =\lim_{x\to0}\frac{\sin^2x}{x^2}・\frac{1}{1+\cos x}\)
\(\displaystyle =\frac{1}{2}\)

Question 20

(20)\(\displaystyle \lim_{x\to0}\frac{x\sin x}{\cos x-1}\)

Answer

\(\displaystyle =\lim_{x\to0}\frac{x\sin x(\cos x+1)}{\cos^2x-1}\)
\(\displaystyle =\lim_{x\to0}\frac{x\sin x(\cos x+1)}{-\sin^2x}\)
\(\displaystyle =\lim_{x\to0}-\frac{x}{\sin x}・(\cos x+1)\)
\(=-2\)