{
  "index": "1956-A-1",
  "type": "ANA",
  "tag": [
    "ANA",
    "ALG"
  ],
  "difficulty": "",
  "question": "1. Evaluate\n\\[\n\\lim _{x \\rightarrow \\infty}\\left[\\frac{1}{x} \\frac{a^{x}-1}{a-1}\\right]^{1 / x}\n\\]\nwhere \\( a>0, a \\neq 1 \\).",
  "solution": "Solution. Let\n\\[\nf(x)=\\left[\\frac{1}{x} \\frac{a^{x}-1}{a-1}\\right]^{1 / x}\n\\]\n\nThen for \\( x>0 \\) and \\( a>1 \\), we have\n\\[\n\\log f(x)=-\\frac{\\log x}{x}-\\frac{\\log (a-1)}{x}+\\frac{\\log \\left(a^{x}-1\\right)}{x}\n\\]\n\nAs \\( x \\rightarrow+\\infty \\),\n\\[\n\\frac{\\log x}{x} \\rightarrow 0, \\frac{\\log (a-1)}{x} \\rightarrow 0\n\\]\nand\n\\[\n\\frac{\\log \\left(a^{x}-1\\right)}{x}=\\frac{\\log \\left(1-a^{-x}\\right)}{x}+\\log a \\rightarrow \\log a .\n\\]\n\nHence \\( \\log f(x) \\rightarrow \\log a \\).\nOn the other hand, if \\( 0<a<1 \\) and \\( x>0 \\), then\n\\[\n\\log f(x)=-\\frac{\\log x}{x}-\\frac{\\log (1-a)}{x}+\\frac{\\log \\left(1-a^{x}\\right)}{x}\n\\]\nand it is clear that all three terms approach zero as \\( x \\rightarrow+\\infty \\).\nSince exp is a continuous function, we have\n\\[\n\\lim _{x \\rightarrow+\\infty} f(x)=\\exp \\lim _{x \\rightarrow+\\infty} \\log f(x)=\\left\\{\\begin{array}{ll}\n\\exp \\log a=a & \\text { if } a>1 \\\\\n\\exp 0=1 & \\text { if } 0<a<1\n\\end{array}\\right.\n\\]\n\nMore concisely, \\( \\lim _{x \\rightarrow+\\infty} f(x)=\\max (a, 1) \\).\nRemark. It is of some interest to see what happens if \\( \\lim _{x-\\infty} \\) is replaced by \\( \\lim _{x-\\infty} \\). To do this, note that for \\( x<0 \\) and \\( a>1 \\), we have\n\\[\n\\log f(x)=-\\frac{\\log |x|}{x}-\\frac{\\log (a-1)}{x}+\\frac{\\log \\left(1-a^{x}\\right)}{x}\n\\]\nand all three terms have limit zero as \\( x \\rightarrow-\\infty \\); since in this case \\( 1-a^{x}-1 \\).\n\nFor \\( x<0 \\) and \\( 0<a<1 \\), on the other hand,\n\\[\n\\log f(x)=-\\frac{\\log |x|}{x}-\\frac{\\log (1-a)}{x}+\\frac{\\log \\left(1-a^{-x}\\right)}{x}+\\log a\n\\]\nso \\( \\lim _{x--\\infty} f(x)=\\log a \\).\nHence we have\n\\[\n\\lim _{x \\rightarrow-\\infty} f(x)=\\left\\{\\begin{array}{ll}\n1 & \\text { if } a>1 \\\\\na & \\text { if } 0<a<1\n\\end{array}\\right.\n\\]\n\nIn other words, \\( \\lim _{x--\\infty} f(x)=\\min (a, 1) \\). In particular \\( \\lim _{x-\\infty} f(x) \\neq \\) \\( \\lim _{x \\rightarrow-\\infty} f(x) \\), so that \\( \\lim _{|x|-\\infty} f(x) \\) does not exist.",
  "vars": [
    "x"
  ],
  "params": [
    "a"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "x": "inputvar",
        "a": "baseconst"
      },
      "question": "1. Evaluate\n\\[\n\\lim _{inputvar \\rightarrow \\infty}\\left[\\frac{1}{inputvar} \\frac{baseconst^{inputvar}-1}{baseconst-1}\\right]^{1 / inputvar}\n\\]\nwhere \\( baseconst>0, baseconst \\neq 1 \\).",
      "solution": "Solution. Let\n\\[\nf(inputvar)=\\left[\\frac{1}{inputvar} \\frac{baseconst^{inputvar}-1}{baseconst-1}\\right]^{1 / inputvar}\n\\]\n\nThen for \\( inputvar>0 \\) and \\( baseconst>1 \\), we have\n\\[\n\\log f(inputvar)=-\\frac{\\log inputvar}{inputvar}-\\frac{\\log (baseconst-1)}{inputvar}+\\frac{\\log \\left(baseconst^{inputvar}-1\\right)}{inputvar}\n\\]\n\nAs \\( inputvar \\rightarrow+\\infty \\),\n\\[\n\\frac{\\log inputvar}{inputvar} \\rightarrow 0, \\frac{\\log (baseconst-1)}{inputvar} \\rightarrow 0\n\\]\nand\n\\[\n\\frac{\\log \\left(baseconst^{inputvar}-1\\right)}{inputvar}=\\frac{\\log \\left(1-baseconst^{-inputvar}\\right)}{inputvar}+\\log baseconst \\rightarrow \\log baseconst .\n\\]\n\nHence \\( \\log f(inputvar) \\rightarrow \\log baseconst \\).\nOn the other hand, if \\( 0<baseconst<1 \\) and \\( inputvar>0 \\), then\n\\[\n\\log f(inputvar)=-\\frac{\\log inputvar}{inputvar}-\\frac{\\log (1-baseconst)}{inputvar}+\\frac{\\log \\left(1-baseconst^{inputvar}\\right)}{inputvar}\n\\]\nand it is clear that all three terms approach zero as \\( inputvar \\rightarrow+\\infty \\).\nSince exp is a continuous function, we have\n\\[\n\\lim _{inputvar \\rightarrow+\\infty} f(inputvar)=\\exp \\lim _{inputvar \\rightarrow+\\infty} \\log f(inputvar)=\\left\\{\\begin{array}{ll}\n\\exp \\log baseconst=baseconst & \\text { if } baseconst>1 \\\\\n\\exp 0=1 & \\text { if } 0<baseconst<1\n\\end{array}\\right.\n\\]\n\nMore concisely, \\( \\lim _{inputvar \\rightarrow+\\infty} f(inputvar)=\\max (baseconst, 1) \\).\nRemark. It is of some interest to see what happens if \\( \\lim _{inputvar-\\infty} \\) is replaced by \\( \\lim _{inputvar-\\infty} \\). To do this, note that for \\( inputvar<0 \\) and \\( baseconst>1 \\), we have\n\\[\n\\log f(inputvar)=-\\frac{\\log |inputvar|}{inputvar}-\\frac{\\log (baseconst-1)}{inputvar}+\\frac{\\log \\left(1-baseconst^{inputvar}\\right)}{inputvar}\n\\]\nand all three terms have limit zero as \\( inputvar \\rightarrow-\\infty \\); since in this case \\( 1-baseconst^{inputvar}-1 \\).\n\nFor \\( inputvar<0 \\) and \\( 0<baseconst<1 \\), on the other hand,\n\\[\n\\log f(inputvar)=-\\frac{\\log |inputvar|}{inputvar}-\\frac{\\log (1-baseconst)}{inputvar}+\\frac{\\log \\left(1-baseconst^{-inputvar}\\right)}{inputvar}+\\log baseconst\n\\]\nso \\( \\lim _{inputvar--\\infty} f(inputvar)=\\log baseconst \\).\nHence we have\n\\[\n\\lim _{inputvar \\rightarrow-\\infty} f(inputvar)=\\left\\{\\begin{array}{ll}\n1 & \\text { if } baseconst>1 \\\\\nbaseconst & \\text { if } 0<baseconst<1\n\\end{array}\\right.\n\\]\n\nIn other words, \\( \\lim _{inputvar--\\infty} f(inputvar)=\\min (baseconst, 1) \\). In particular \\( \\lim _{inputvar-\\infty} f(inputvar) \\neq \\) \\( \\lim _{inputvar \\rightarrow-\\infty} f(inputvar) \\), so that \\( \\lim _{|inputvar|-\\infty} f(inputvar) \\) does not exist."
    },
    "descriptive_long_confusing": {
      "map": {
        "x": "blueberry",
        "a": "cloudburst"
      },
      "question": "1. Evaluate\n\\[\n\\lim _{blueberry \\rightarrow \\infty}\\left[\\frac{1}{blueberry} \\frac{cloudburst^{blueberry}-1}{cloudburst-1}\\right]^{1 / blueberry}\n\\]\nwhere \\( cloudburst>0, cloudburst \\neq 1 \\).",
      "solution": "Solution. Let\n\\[\nf(blueberry)=\\left[\\frac{1}{blueberry} \\frac{cloudburst^{blueberry}-1}{cloudburst-1}\\right]^{1 / blueberry}\n\\]\n\nThen for \\( blueberry>0 \\) and \\( cloudburst>1 \\), we have\n\\[\n\\log f(blueberry)=-\\frac{\\log blueberry}{blueberry}-\\frac{\\log (cloudburst-1)}{blueberry}+\\frac{\\log \\left(cloudburst^{blueberry}-1\\right)}{blueberry}\n\\]\n\nAs \\( blueberry \\rightarrow+\\infty \\),\n\\[\n\\frac{\\log blueberry}{blueberry} \\rightarrow 0, \\frac{\\log (cloudburst-1)}{blueberry} \\rightarrow 0\n\\]\nand\n\\[\n\\frac{\\log \\left(cloudburst^{blueberry}-1\\right)}{blueberry}=\\frac{\\log \\left(1-cloudburst^{-blueberry}\\right)}{blueberry}+\\log cloudburst \\rightarrow \\log cloudburst .\n\\]\n\nHence \\( \\log f(blueberry) \\rightarrow \\log cloudburst \\).\nOn the other hand, if \\( 0<cloudburst<1 \\) and \\( blueberry>0 \\), then\n\\[\n\\log f(blueberry)=-\\frac{\\log blueberry}{blueberry}-\\frac{\\log (1-cloudburst)}{blueberry}+\\frac{\\log \\left(1-cloudburst^{blueberry}\\right)}{blueberry}\n\\]\nand it is clear that all three terms approach zero as \\( blueberry \\rightarrow+\\infty \\).\nSince exp is a continuous function, we have\n\\[\n\\lim _{blueberry \\rightarrow+\\infty} f(blueberry)=\\exp \\lim _{blueberry \\rightarrow+\\infty} \\log f(blueberry)=\\left\\{\\begin{array}{ll}\n\\exp \\log cloudburst=cloudburst & \\text { if } cloudburst>1 \\\\\n\\exp 0=1 & \\text { if } 0<cloudburst<1\n\\end{array}\\right.\n\\]\n\nMore concisely, \\( \\lim _{blueberry \\rightarrow+\\infty} f(blueberry)=\\max (cloudburst, 1) \\).\nRemark. It is of some interest to see what happens if \\( \\lim _{blueberry-\\infty} \\) is replaced by \\( \\lim _{blueberry-\\infty} \\). To do this, note that for \\( blueberry<0 \\) and \\( cloudburst>1 \\), we have\n\\[\n\\log f(blueberry)=-\\frac{\\log |blueberry|}{blueberry}-\\frac{\\log (cloudburst-1)}{blueberry}+\\frac{\\log \\left(1-cloudburst^{blueberry}\\right)}{blueberry}\n\\]\nand all three terms have limit zero as \\( blueberry \\rightarrow-\\infty \\); since in this case \\( 1-cloudburst^{blueberry}-1 \\).\n\nFor \\( blueberry<0 \\) and \\( 0<cloudburst<1 \\), on the other hand,\n\\[\n\\log f(blueberry)=-\\frac{\\log |blueberry|}{blueberry}-\\frac{\\log (1-cloudburst)}{blueberry}+\\frac{\\log \\left(1-cloudburst^{-blueberry}\\right)}{blueberry}+\\log cloudburst\n\\]\nso \\( \\lim _{blueberry--\\infty} f(blueberry)=\\log cloudburst \\).\nHence we have\n\\[\n\\lim _{blueberry \\rightarrow-\\infty} f(blueberry)=\\left\\{\\begin{array}{ll}\n1 & \\text { if } cloudburst>1 \\\\\ncloudburst & \\text { if } 0<cloudburst<1\n\\end{array}\\right.\n\\]\n\nIn other words, \\( \\lim _{blueberry--\\infty} f(blueberry)=\\min (cloudburst, 1) \\). In particular \\( \\lim _{blueberry-\\infty} f(blueberry) \\neq \\) \\( \\lim _{blueberry \\rightarrow-\\infty} f(blueberry) \\), so that \\( \\lim _{|blueberry|-\\infty} f(blueberry) \\) does not exist."
    },
    "descriptive_long_misleading": {
      "map": {
        "x": "constant",
        "a": "changing"
      },
      "question": "1. Evaluate\n\\[\n\\lim _{constant \\rightarrow \\infty}\\left[\\frac{1}{constant} \\frac{changing^{constant}-1}{changing-1}\\right]^{1 / constant}\\]\nwhere \\( changing>0, changing \\neq 1 \\).",
      "solution": "Solution. Let\n\\[\nf(constant)=\\left[\\frac{1}{constant} \\frac{changing^{constant}-1}{changing-1}\\right]^{1 / constant}\n\\]\n\nThen for \\( constant>0 \\) and \\( changing>1 \\), we have\n\\[\n\\log f(constant)=-\\frac{\\log constant}{constant}-\\frac{\\log (changing-1)}{constant}+\\frac{\\log \\left(changing^{constant}-1\\right)}{constant}\n\\]\n\nAs \\( constant \\rightarrow+\\infty \\),\n\\[\n\\frac{\\log constant}{constant} \\rightarrow 0, \\frac{\\log (changing-1)}{constant} \\rightarrow 0\n\\]\nand\n\\[\n\\frac{\\log \\left(changing^{constant}-1\\right)}{constant}=\\frac{\\log \\left(1-changing^{-constant}\\right)}{constant}+\\log changing \\rightarrow \\log changing .\n\\]\n\nHence \\( \\log f(constant) \\rightarrow \\log changing \\).\nOn the other hand, if \\( 0<changing<1 \\) and \\( constant>0 \\), then\n\\[\n\\log f(constant)=-\\frac{\\log constant}{constant}-\\frac{\\log (1-changing)}{constant}+\\frac{\\log \\left(1-changing^{constant}\\right)}{constant}\n\\]\nand it is clear that all three terms approach zero as \\( constant \\rightarrow+\\infty \\).\nSince exp is a continuous function, we have\n\\[\n\\lim _{constant \\rightarrow+\\infty} f(constant)=\\exp \\lim _{constant \\rightarrow+\\infty} \\log f(constant)=\\left\\{\\begin{array}{ll}\n\\exp \\log changing=changing & \\text { if } changing>1 \\\\\n\\exp 0=1 & \\text { if } 0<changing<1\n\\end{array}\\right.\n\\]\n\nMore concisely, \\( \\lim _{constant \\rightarrow+\\infty} f(constant)=\\max (changing, 1) \\).\nRemark. It is of some interest to see what happens if \\( \\lim _{constant-\\infty} \\) is replaced by \\( \\lim _{constant-\\infty} \\). To do this, note that for \\( constant<0 \\) and \\( changing>1 \\), we have\n\\[\n\\log f(constant)=-\\frac{\\log |constant|}{constant}-\\frac{\\log (changing-1)}{constant}+\\frac{\\log \\left(1-changing^{constant}\\right)}{constant}\n\\]\nand all three terms have limit zero as \\( constant \\rightarrow-\\infty \\); since in this case \\( 1-changing^{constant}-1 \\).\n\nFor \\( constant<0 \\) and \\( 0<changing<1 \\), on the other hand,\n\\[\n\\log f(constant)=-\\frac{\\log |constant|}{constant}-\\frac{\\log (1-changing)}{constant}+\\frac{\\log \\left(1-changing^{-constant}\\right)}{constant}+\\log changing\n\\]\nso \\( \\lim _{constant--\\infty} f(constant)=\\log changing \\).\nHence we have\n\\[\n\\lim _{constant \\rightarrow-\\infty} f(constant)=\\left\\{\\begin{array}{ll}\n1 & \\text { if } changing>1 \\\\\nchanging & \\text { if } 0<changing<1\n\\end{array}\\right.\n\\]\n\nIn other words, \\( \\lim _{constant--\\infty} f(constant)=\\min (changing, 1) \\). In particular \\( \\lim _{constant-\\infty} f(constant) \\neq \\) \\( \\lim _{constant \\rightarrow-\\infty} f(constant) \\), so that \\( \\lim _{|constant|-\\infty} f(constant) \\) does not exist."
    },
    "garbled_string": {
      "map": {
        "x": "qzxwvtnp",
        "a": "hjgrksla"
      },
      "question": "Evaluate\n\\[\n\\lim _{qzxwvtnp \\rightarrow \\infty}\\left[\\frac{1}{qzxwvtnp} \\frac{hjgrksla^{qzxwvtnp}-1}{hjgrksla-1}\\right]^{1 / qzxwvtnp}\n\\]\nwhere \\( hjgrksla>0, hjgrksla \\neq 1 \\).",
      "solution": "Solution. Let\n\\[\nf(qzxwvtnp)=\\left[\\frac{1}{qzxwvtnp} \\frac{hjgrksla^{qzxwvtnp}-1}{hjgrksla-1}\\right]^{1 / qzxwvtnp}\n\\]\n\nThen for \\( qzxwvtnp>0 \\) and \\( hjgrksla>1 \\), we have\n\\[\n\\log f(qzxwvtnp)=-\\frac{\\log qzxwvtnp}{qzxwvtnp}-\\frac{\\log (hjgrksla-1)}{qzxwvtnp}+\\frac{\\log \\left(hjgrksla^{qzxwvtnp}-1\\right)}{qzxwvtnp}\n\\]\n\nAs \\( qzxwvtnp \\rightarrow+\\infty \\),\n\\[\n\\frac{\\log qzxwvtnp}{qzxwvtnp} \\rightarrow 0, \\frac{\\log (hjgrksla-1)}{qzxwvtnp} \\rightarrow 0\n\\]\nand\n\\[\n\\frac{\\log \\left(hjgrksla^{qzxwvtnp}-1\\right)}{qzxwvtnp}=\\frac{\\log \\left(1-hjgrksla^{-qzxwvtnp}\\right)}{qzxwvtnp}+\\log hjgrksla \\rightarrow \\log hjgrksla .\n\\]\n\nHence \\( \\log f(qzxwvtnp) \\rightarrow \\log hjgrksla \\).\nOn the other hand, if \\( 0<hjgrksla<1 \\) and \\( qzxwvtnp>0 \\), then\n\\[\n\\log f(qzxwvtnp)=-\\frac{\\log qzxwvtnp}{qzxwvtnp}-\\frac{\\log (1-hjgrksla)}{qzxwvtnp}+\\frac{\\log \\left(1-hjgrksla^{qzxwvtnp}\\right)}{qzxwvtnp}\n\\]\nand it is clear that all three terms approach zero as \\( qzxwvtnp \\rightarrow+\\infty \\).\nSince exp is a continuous function, we have\n\\[\n\\lim _{qzxwvtnp \\rightarrow+\\infty} f(qzxwvtnp)=\\exp \\lim _{qzxwvtnp \\rightarrow+\\infty} \\log f(qzxwvtnp)=\\left\\{\\begin{array}{ll}\n\\exp \\log hjgrksla=hjgrksla & \\text { if } hjgrksla>1 \\\\\n\\exp 0=1 & \\text { if } 0<hjgrksla<1\n\\end{array}\\right.\n\\]\n\nMore concisely, \\( \\lim _{qzxwvtnp \\rightarrow+\\infty} f(qzxwvtnp)=\\max (hjgrksla, 1) \\).\nRemark. It is of some interest to see what happens if \\( \\lim _{qzxwvtnp-\\infty} \\) is replaced by \\( \\lim _{qzxwvtnp-\\infty} \\). To do this, note that for \\( qzxwvtnp<0 \\) and \\( hjgrksla>1 \\), we have\n\\[\n\\log f(qzxwvtnp)=-\\frac{\\log |qzxwvtnp|}{qzxwvtnp}-\\frac{\\log (hjgrksla-1)}{qzxwvtnp}+\\frac{\\log \\left(1-hjgrksla^{qzxwvtnp}\\right)}{qzxwvtnp}\n\\]\nand all three terms have limit zero as \\( qzxwvtnp \\rightarrow-\\infty \\); since in this case \\( 1-hjgrksla^{qzxwvtnp}-1 \\).\n\nFor \\( qzxwvtnp<0 \\) and \\( 0<hjgrksla<1 \\), on the other hand,\n\\[\n\\log f(qzxwvtnp)=-\\frac{\\log |qzxwvtnp|}{qzxwvtnp}-\\frac{\\log (1-hjgrksla)}{qzxwvtnp}+\\frac{\\log \\left(1-hjgrksla^{-qzxwvtnp}\\right)}{qzxwvtnp}+\\log hjgrksla\n\\]\nso \\( \\lim _{qzxwvtnp--\\infty} f(qzxwvtnp)=\\log hjgrksla \\).\nHence we have\n\\[\n\\lim _{qzxwvtnp \\rightarrow-\\infty} f(qzxwvtnp)=\\left\\{\\begin{array}{ll}\n1 & \\text { if } hjgrksla>1 \\\\\n hjgrksla & \\text { if } 0<hjgrksla<1\n\\end{array}\\right.\n\\]\n\nIn other words, \\( \\lim _{qzxwvtnp--\\infty} f(qzxwvtnp)=\\min (hjgrksla, 1) \\). In particular \\( \\lim _{qzxwvtnp-\\infty} f(qzxwvtnp) \\neq \\) \\( \\lim _{qzxwvtnp \\rightarrow-\\infty} f(qzxwvtnp) \\), so that \\( \\lim _{|qzxwvtnp|-\\infty} f(qzxwvtnp) \\) does not exist."
    },
    "kernel_variant": {
      "question": "For a positive parameter a with a≠1, set  \n  F_a(x)=⎡  2√x            |a^{√x}−ln x| ⎤^{\\,10/√x},  x>1.  \n        (x²+1)(a²+4)  \nEvaluate the limit  \n  L(a)=lim_{x→∞} F_a(x)  \nas an explicit function of a.\n\n──────────────────",
      "solution": "Put  \n f(x)=  2√x ·|a^{√x}−ln x| /(x²+1)(a²+4), so F_a(x)=f(x)^{10/√x}.  \n\nStep 1 – logarithm.  \n ln F_a(x)=10/√x·[ln 2+½ ln x−ln(x²+1)−ln(a²+4)+ln|a^{√x}−ln x|].\n\nStep 2 – dominant factor inside |·|.  \n\n(i) a>1.  For large x, a^{√x}≫ln x, hence |a^{√x}−ln x|=a^{√x}−ln x>0 and  \n\n ln|a^{√x}−ln x|=ln(a^{√x})+ln(1−ln x·a^{−√x})  \n           =√x ln a+o(1).  \n\n(ii) 0<a<1.  Now a^{√x}→0 while ln x→∞, so |a^{√x}−ln x|=ln x−a^{√x}. Consequently  \n\n ln|a^{√x}−ln x|=ln ln x+ln(1−a^{√x}/ln x)=ln ln x+o(1).  \n\nStep 3 – the small baggage.  \nBecause ln x/√x→0 and every constant divided by √x vanishes, we have  \n\n 10/√x·[ln 2+½ ln x−ln(x²+1)−ln(a²+4)]=O(ln x/√x)→0.  \n\nStep 4 – assemble the leading term.  \n\n(i) a>1: ln F_a(x)=10/√x·(√x ln a+o(1))+o(1)=10 ln a.  \n\n(ii) 0<a<1: ln F_a(x)=10/√x·(ln ln x+o(1))+o(1)→0.  \n\nStep 5 – back to the original scale.  \n L(a)=exp lim_{x→∞} ln F_a(x)=  \n  { a^{10}, a>1;  \n   1,   0<a<1. }  \n\nEquivalently, L(a)=max{1,a^{10}}.  \n\nRemark.  The borderline case a=1 was excluded because the expression |a^{√x}−ln x| would then simplify to |1−ln x|, changing the leading behaviour; one checks separately that F_1(x)→1.\n\n──────────────────",
      "_replacement_note": {
        "replaced_at": "2025-07-05T22:17:12.134828",
        "reason": "Original kernel variant was too easy compared to the original problem"
      }
    }
  },
  "checked": true,
  "problem_type": "calculation"
}