H2-LIMIT-2026-05-22-D4-BULK001어려움함수의 극한과 연속
수능 수학 전문가의 극한과 연속 고난도 문제
함수의 극한과 연속 단원에서 출제될 수 있는 고난도 문항입니다. 연속성과 무한대 극한을 종합적으로 묻습니다.
2026학년도 수능고등학교 2학년
문제
함수 가 모든 실수 에 대하여 연속이고, 다음과 같이 정의될 때, 의 값을 구하시오.
\frac{\sqrt{ax^2+b}-3}{x-2} & (x \neq 2) \\ k & (x=2) \end{cases}$$ 단, $\lim_{x \to \infty} f(x) = c$ (c는 유한한 값) 이며, $k=2$ 이다. <svg viewBox="0 0 400 300" xmlns="http://www.w3.org/2000/svg"> <!-- Define styles for consistent appearance --> <defs> <style> .background { fill: #F8F7FF; /* Light lavender background */ } .main-line { stroke: #1F2937; /* Dark gray for axes and function */ stroke-width: 2; fill: none; stroke-linecap: round; stroke-linejoin: round; } .dash-line { stroke: #6C63FF; /* Purple for dashed asymptotes */ stroke-width: 2; fill: none; stroke-dasharray: 5 5; } .point { fill: #1F2937; /* Dark gray for points */ stroke: none; } .label { font-family: 'Noto Sans KR', sans-serif; font-size: 13px; fill: #1F2937; /* Dark gray for labels */ } .axis-label { font-family: 'Noto Sans KR', sans-serif; font-size: 14px; fill: #1F2937; /* Dark gray for axis labels */ } </style> </defs> <!-- Background rectangle --> <rect x="0" y="0" width="400" height="300" class="background"/> <!-- Coordinate System Setup --> <!-- Origin set to (200, 150) in SVG coordinates, with 1 unit = 35 pixels --> <!-- This allows for plotting points around x-axis and y-axis within the viewBox --> <!-- y-coordinates are inverted in SVG (positive y goes down) --> <!-- Axes --> <g class="main-line"> <line x1="25" y1="150" x2="385" y2="150" /> <!-- X-axis --> <line x1="200" y1="20" x2="200" y2="280" /> <!-- Y-axis --> </g> <!-- Axis labels --> <text x="390" y="155" class="axis-label" dominant-baseline="middle">x</text> <text x="205" y="20" class="axis-label" text-anchor="middle">y</text> <text x="190" y="165" class="axis-label">O</text> <!-- Ticks and their labels --> <g class="main-line"> <!-- X-tick for x=2 --> <line x1="270" y1="145" x2="270" y2="155" /> <text x="270" y="165" class="label" text-anchor="middle">2</text> <!-- Y-tick for y=2 --> <line x1="195" y1="80" x2="205" y2="80" /> <text x="185" y="80" class="label" text-anchor="end" dominant-baseline="middle">2</text> </g> <!-- Horizontal Asymptote y = c = sqrt(3) --> <!-- sqrt(3) is approximately 1.732 --> <!-- SVG y-coordinate for y=sqrt(3): 150 - 1.732 * 35 = 150 - 60.62 = 89.38 --> <line x1="25" y1="89.38" x2="385" y2="89.38" class="dash-line" /> <text x="380" y="89.38" class="label" text-anchor="end" dy="-5">y = c = √3</text> <!-- Horizontal Asymptote y = -sqrt(3) (for completeness as x -> -infinity) --> <!-- SVG y-coordinate for y=-sqrt(3): 150 - (-1.732) * 35 = 150 + 60.62 = 210.62 --> <line x1="25" y1="210.62" x2="385" y2="210.62" class="dash-line" /> <!-- Function curve f(x) --> <!-- This path consists of two branches, one for x >= 1 and one for x <= -1, approximated by line segments connecting calculated points. --> <path class="main-line" d=" M 235 45 <!-- Point (1,3) --> L 270 80 <!-- Point (2,2) - continuity point --> L 305 83.5 <!-- Point (3, 1.90) --> L 340 85.25 <!-- Point (4, 1.85) --> L 375 86.3 <!-- Point (5, 1.82) - approaches y=sqrt(3) --> "/> <path class="main-line" d=" M 165 115 <!-- Point (-1,1) --> L 130 150 <!-- Point (-2,0) --> L 95 163.3 <!-- Point (-3, -0.38) --> L 60 171.7 <!-- Point (-4, -0.62) --> L 25 177.3 <!-- Point (-5, -0.78) - approaches y=-sqrt(3) --> "/> <!-- Point P(2,2) where continuity is guaranteed --> <circle cx="270" cy="80" r="4" class="point" /> <text x="270" y="80" class="label" dx="8" dy="-5">P(2,2)</text> </svg>🔐
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#함수의 극한#함수의 연속#극한값 계산#미정계수 결정#무리함수#수학II#함수의 극한과 연속