The Gap Between Two Exponentials
Difficulty: Medium · Topics: Calculus, Limits, Exponentials
Ikki ko‘rsatkichli funksiya orasidagi masofa
The graphs of y=4^x and y=3^x meet the line x=a (a 0) at the points P and Q respectively.
graph "x":[-1.3,1.6],"y":[-0.35,2.6],"xTicks":[],"yTicks":[1],"label":"The graphs of y = 4ˣ and y = 3ˣ meeting the line x = a at P and Q","items":["type":"curve","points":[[-1.2,0.1895],[-1.15,0.2031],[-1.1,0.2176],[-1.05,0.2333],[-1,0.25],[-0.95,0.2679],[-0.9,0.2872],[-0.85,0.3078],[-0.8,0.3299],[-0.75,0.3536],[-0.7,0.3789],[-0.65,0.4061],[-0.6,0.4353],[-0.55,0.4665],[-0.5,0.5],[-0.45,0.5359],[-0.4,0.5743],[-0.35,0.6156],[-0.3,0.6598],[-0.25,0.7071],[-0.2,0.7579],[-0.15,0.8123],[-0.1,0.8706],[-0.05,0.933],[0,1],[0.05,1.0718],[0.1,1.1487],[0.15,1.2311],[0.2,1.3195],[0.25,1.4142],[0.3,1.5157],[0.35,1.6245],[0.4,1.7411],[0.45,1.8661],[0.5,2],[0.55,2.1435],[0.6,2.2974],[0.65,2.4623],[0.7,2.639],[0.75,2.8284],[0.8,3.0314],[0.85,3.249],[0.9,3.4822],[0.95,3.7321],[1,4],[1.05,4.2871],[1.1,4.5948],[1.15,4.9246],[1.2,5.278]],"ink":false,"type":"curve","points":[[-1.2,0.2676],[-1.15,0.2827],[-1.1,0.2987],[-1.05,0.3155],[-1,0.3333],[-0.95,0.3522],[-0.9,0.372],[-0.85,0.393],[-0.8,0.4152],[-0.75,0.4387],[-0.7,0.4635],[-0.65,0.4896],[-0.6,0.5173],[-0.55,0.5465],[-0.5,0.5774],[-0.45,0.61],[-0.4,0.6444],[-0.35,0.6808],[-0.3,0.7192],[-0.25,0.7598],[-0.2,0.8027],[-0.15,0.8481],[-0.1,0.896],[-0.05,0.9466],[0,1],[0.05,1.0565],[0.1,1.1161],[0.15,1.1791],[0.2,1.2457],[0.25,1.3161],[0.3,1.3904],[0.35,1.4689],[0.4,1.5518],[0.45,1.6395],[0.5,1.7321],[0.55,1.8299],[0.6,1.9332],[0.65,2.0423],[0.7,2.1577],[0.75,2.2795],[0.8,2.4082],[0.85,2.5442],[0.9,2.6879],[0.95,2.8397],[1,3],[1.05,3.1694],[1.1,3.3484],[1.15,3.5374],[1.2,3.7372]],"ink":false,"type":"line","from":[0.62,-0.3],"to":[0.62,2.5],"ink":true,"dash":true,"type":"point","at":[0.62,2.362],"label":"P","anchor":"nw","ink":false,"type":"point","at":[0.62,1.9761],"label":"Q","anchor":"se","ink":false,"type":"label","at":[1.15,2.35],"text":"y = 4ˣ","type":"label","at":[1.45,1.55],"text":"y = 3ˣ","type":"label","at":[0.95,-0.2],"text":"x = a"]
What is
y=4^x va y=3^x grafiklari x=a (a 0) to‘g‘ri chiziqni mos ravishda P va Q nuqtalarda kesadi.
graph "x":[-1.3,1.6],"y":[-0.35,2.6],"xTicks":[],"yTicks":[1],"label":"The graphs of y = 4ˣ and y = 3ˣ meeting the line x = a at P and Q","items":["type":"curve","points":[[-1.2,0.1895],[-1.15,0.2031],[-1.1,0.2176],[-1.05,0.2333],[-1,0.25],[-0.95,0.2679],[-0.9,0.2872],[-0.85,0.3078],[-0.8,0.3299],[-0.75,0.3536],[-0.7,0.3789],[-0.65,0.4061],[-0.6,0.4353],[-0.55,0.4665],[-0.5,0.5],[-0.45,0.5359],[-0.4,0.5743],[-0.35,0.6156],[-0.3,0.6598],[-0.25,0.7071],[-0.2,0.7579],[-0.15,0.8123],[-0.1,0.8706],[-0.05,0.933],[0,1],[0.05,1.0718],[0.1,1.1487],[0.15,1.2311],[0.2,1.3195],[0.25,1.4142],[0.3,1.5157],[0.35,1.6245],[0.4,1.7411],[0.45,1.8661],[0.5,2],[0.55,2.1435],[0.6,2.2974],[0.65,2.4623],[0.7,2.639],[0.75,2.8284],[0.8,3.0314],[0.85,3.249],[0.9,3.4822],[0.95,3.7321],[1,4],[1.05,4.2871],[1.1,4.5948],[1.15,4.9246],[1.2,5.278]],"ink":false,"type":"curve","points":[[-1.2,0.2676],[-1.15,0.2827],[-1.1,0.2987],[-1.05,0.3155],[-1,0.3333],[-0.95,0.3522],[-0.9,0.372],[-0.85,0.393],[-0.8,0.4152],[-0.75,0.4387],[-0.7,0.4635],[-0.65,0.4896],[-0.6,0.5173],[-0.55,0.5465],[-0.5,0.5774],[-0.45,0.61],[-0.4,0.6444],[-0.35,0.6808],[-0.3,0.7192],[-0.25,0.7598],[-0.2,0.8027],[-0.15,0.8481],[-0.1,0.896],[-0.05,0.9466],[0,1],[0.05,1.0565],[0.1,1.1161],[0.15,1.1791],[0.2,1.2457],[0.25,1.3161],[0.3,1.3904],[0.35,1.4689],[0.4,1.5518],[0.45,1.6395],[0.5,1.7321],[0.55,1.8299],[0.6,1.9332],[0.65,2.0423],[0.7,2.1577],[0.75,2.2795],[0.8,2.4082],[0.85,2.5442],[0.9,2.6879],[0.95,2.8397],[1,3],[1.05,3.1694],[1.1,3.3484],[1.15,3.5374],[1.2,3.7372]],"ink":false,"type":"line","from":[0.62,-0.3],"to":[0.62,2.5],"ink":true,"dash":true,"type":"point","at":[0.62,2.362],"label":"P","anchor":"nw","ink":false,"type":"point","at":[0.62,1.9761],"label":"Q","anchor":"se","ink":false,"type":"label","at":[1.15,2.35],"text":"y = 4ˣ","type":"label","at":[1.45,1.55],"text":"y = 3ˣ","type":"label","at":[0.95,-0.2],"text":"x = a"]
lim a