Question
- 1
- 0
![1 half](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABIAAAAjCAYAAACZ6FpfAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAXQ/cXWQAAAJFJREFUeNpjYMAP1IC4FogvMFAIFgNxGhD/Z6ASGDVo1KBBYdB/LHgUDCbwn0w8CgYLsAbiNUD8CYh/QaujaHIMOgjEkUDMA+VrAfFRqBjFQB6IL1HLyz+oYYgl1HsUAQ4gPgmNBLKBIBBvAGI3SgxRghqiQokhGkA8G4i5KDFEHIhXATELpYG7BeoimhZyWAEAI3M1I31CbrEAAABidEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1mcmFjPjxtbj4xPC9tbj48bW4+MjwvbW4+PC9tZnJhYz48L21hdGg+ND6jrQAAAABJRU5ErkJggg==)
- 2
Hint:
If the function has a square root in it and Substitution yields ![0 over 0](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABIAAAAjCAYAAACZ6FpfAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAXQ/cXWQAAAJtJREFUeNpjYMAN+IB4GhCfhOKZUDGSwV4g9kPi+0DFSAKhQDwJi/gEII4kxaA1QOyGRdwRKkc0eAfEbFjEQWIvSTHoDx65X9Qy6AcpBr3E4TUOUr0GClAPLOJupAZ2EBDPxiI+n9ToB4H9QFwAxCxQb1YD8UFyUrYgEM+FBu43aBbhYRgFgwv8JxOPgsEERquj0epotDqiXnUEAHl3QY/fxbExAAAAYnRFWHRNYXRoTUwAPG1hdGggeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzE5OTgvTWF0aC9NYXRoTUwiPjxtZnJhYz48bW4+MDwvbW4+PG1uPjA8L21uPjwvbWZyYWM+PC9tYXRoPt4dSgEAAAAASUVORK5CYII=)
, 0 divided by 0, then multiply the numerator and the denominator by
1=![Error converting from MathML to accessible text.](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAg8AAAAoCAYAAAB5ExkFAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAZpE86XgAAD85JREFUeNrtXXFkXUkXH09ERZVYERVVIlZVVPhU1YpV1qqqiPxT/WOtWqo+sVaVqIpYVVZ9VlUsqz4RUcuq6B+1SlXEqs9Sq6oiwlpVFRUiKqoq9LtHztW7kzNzz8yZ+959zfkxvDdv7txzfufMufPOnXvHGIXi48e/s/KD0qBQtBw/4HhUKBQVopGVXqVBhCNZ+Z/SoFDUBo9wXCp2Afqzcj0rT7LyJivPs3I1K90Vn/cNXkB3I05mZR3Ls6x0KH/RgWpIh/Cu95OPQcePxU7/ysrvNeCpCj415hZwKSv3sjJcIOVAVr7LykyF5x3IytMW6z6AF59mAyYKLwsXvcMRMlbNX6u4CcFQG8jYaj+uwzhrhv5PVYdaje/fcRJRNU9VxMdWxdy2uvZNZOXWLg66Y1m53aLz/iKUcTSgj3biJgSQLRvXuUNb2KpKVD0W1GfC8a1pzjqkKuJjq2Ju29j/06z8Zfjp8p6s/Dcrm1l5m5XZrOyx2kxhJqMXsxaQ4llHRzJE2wnPd199F1441lCWO1mZttpBFuX7rLzKyjuzfUvmoNXv+0Kxz9GL/UL/cBvn88Q8TTD68smY9/FJVn5EPV1ch+oi5SaX7UJWHmO7Ces36OMmyr2alVP4ezfqA7q8zsplj5z3zXbGzET6IdfnpDJL+JL6MaVj6rEsHQeU3i7sRRtseMY+l3OufiG6xPpISp+T+kwK2x3PysPAC9dVtIkN6GeQ4ClVfKxDzOVwyrELp12Z/UvxH7N9a4KDbkzXnEPH3IczsOtWu1+RyBkkC9r2oQJdRNtRz3dXPUx2FlHhDiwXs7JltbuHg64b5aD6d51zH+o7gt/htsKzRDzNW4Y7W9Knj5cpTA+OeLiO1UXCDRz7En2s3yH3Y9S9AwPb31nZj7Y9g/UHUB/XolIImp0O2Tl+yPU5icxSviR+TP1WxViWxAuX3hSgrwWr3zwQjkb4KEe/EF0kfp3S5yQ+k8p2nTg+KQwQ9gDuV7BfY01C7kReNzjxsW4x18Up1y4h9huNzTwsoRG56eGLVl0/GrsI+H6eOH4dZ2t22z7Pd1f9ZFauOdrtx89nCSecLczacyxbM/KivpeJIN6ZiKdlHPAcuGRcQZn2lnAdq4uEmxX8d2Eccj80OxfjwqC55ajf7+hry3MOjh9yfU4is4QvqR9TulQxliXxYiwgZk06/iVROnI45+gXoovEr1P5XIrYl8p27zy/2QsP4Z8yLCJ/YbWD+/KHPDxJ42OdYq6PU65dQq5BB2MmDg2cLXHbrhHpkT2YOiy22ySO70BH8dU1iDau+lXC+Pa5F3DGWsSiRZbvnBuY+inWvTb+VbYhPL0J4N4l4yYxg6Z4jdUllhuffvlvXYH1JmDywPVDrs9JZJbyJfFjly6px3IzxkGOF0TgbkT4fIiPhOgS69cpfU4a+1Lazjd5gMerj+LnQZQxv6jlE6OT1kSoETCGOfGxHWJuqF2qGnuslJIN12M3x9BRXd+LaaeFkjrXsXY9VxZ7VksFi2MFh7Xr3xPlUcU8GcexLhkfMriO1UXCjU8/rp25XFG3Lbh+yD2nRGYpXxI/loyfkLHcjHEAGPL45EIiHz2emK9YH5P0J/GZlLYru8ZANuQMfn5gPjyZAZOFPN3/xMo6UHJI4mPdYu6CcEyF2G/RCEDNoiiMEGkwAKyXuGalWGccqdeZkjrXsV9b9ZDSmWOcY836/bOs/EkcMxugb9U8GYdes0xOqfpYXSTc+PQL8REOVw/QtjF9cX1OIrOUL4kfh/iDZCw3YxyEjH0J583iq8wXJf1VEftibOdbMAmAt1DCQtcTWblbqIeFnrCugHqygeJpNmE8aXXMnRGOqRD7zUomD3CP6RKjHaSOfiPSKEvWrBAW6HxDHE/VQ13xfuMNs3PFKqS0Vq1jRwz9qMyc1e6llbqBe6V3rGPgnOMOp/8jgs8Qns4z+3TJCH2cY3Adq4uEG5cf+HQPrc9BParJ9UOuz0lklvIl8WOq7yrGcup44cKoIzDOWXaQcC7hS+LXNxP6nMRnUtpu3OxcpFcErMGYJ7ILY1j/zKp3jWFJfKxTzPXJxrULt50vZrAA98BgoQcs7MjvMYHTncaJxRdYl7/Q6DR+70PjXrH6mzc7F+W46u9adVM4EwJFe/D7NHEsrBx9ZT68XGkIBwbIV3wUBh75+Rn7Az1vE6ROo54UlnC21kBugKPhEj6lPFFwyRjCdYwuEm58+oXIzeGKekkUty+uz0lllvAl9WO77yrGcjPGgcEs6VJh7EOK9h6e61QiH20WX2U+JulP4jMpbVf2kiiw53vi3zZc5LYcWSb7vCniY11irk82rl247Xwxg40BDKCv0ZAbeDG2lRhGQmABzLJjBrZu6GdOqXq7Dgi+j/0vFpSnjoXsw3NsC48xfYmpOnsxyhX8pzKJfbwy/3w05TASDXrb98370fm3UG/uPyQJTxRcMoZwHaOLhBuffiFyc7mChVdHIvri+pxUZglfUj+m+k49lpsxDnL0YjB8i3Y/EenzIfpVwVeZj0n7k/hMCttxX0+9Znau+G/guQ8xzpsiPtYl5pbJxrELt53P/rUGBICVhP3BSl3OW796aqA7yLBqFCmx2zbG6lGTK2ruM7oxliI5YFYJL8C4GHk8zEZ/Mh+ejz5qdj4HXFfcwJmgbh2dHheUV4WiFoB1DueVBkUqfGW2U42bwiCfv5oW+oFHkuBe3mCbcACpvDF1BYVCoVAoFAqFQqFQKBQKhUKhUCgA77Vo0aJFixYtWjxFoVAoFAqFQqFQKBQKhUKhUCgUCoVCoVA0F7DJjr7nQaFoPX7A8ahQKCoEvDSrV2kQYbe9YVKhqDv0DZO7CPAubngzGOxwBi9fgr0jYIOV7orPa+89v5sAO6CtY4Ed5DqUv+hANaRDeNf7yceg48diJ+7eFlXzVAWfGnMLgG25YXe64QIpB8z2rmAzFZ4XNuV62mLdB8zOXRmbgXz3s/yidzhCxqr5axU3IRhqAxlb7cd1GGfN0P+p6lCr8V22q2YqnqqIj62KuW117ZvIyq1dHHThddG3W3TeX4Qyjgb00U7chACyZeNG0Q62qhJVjwX1mXB8a5qzDqmK+NiqmNs29v80K38ZfrocdmaDPcBhXwnYpwK28ra3EJ3CTEYvZi0gxbOOjmSIthOe7776LrxwrKEssI34tNUOsijfm+3taGF7Urglc9Dqt/hiDPscvdgv9A+3cT5PzNMEoy+fjHkfn2TlR9TTxXWoLlJuctlgw6rH2G7C+g36uIlyw06j+Tbw3agP6AJbxV/2yAlbag875Of4IdfnpDJL+JL6MaVj6rEsHQeU3i7k+9tseMY+l3OufiG6xPpISp+T+kwK2x3PysPAC9dVtIkN6GeQ4ClVfKxDzOVwyrELp12Z/UsBu1t+x2zbjemac+iY+3AGdt1q9ysSOYNkQds+VKCLaDvq+e6qh8nOIircgQV26dyy2t3DQdeNclD9u865D/Udwe9wW+FZIp7mLcOdLenTx8sUpgdHPFzH6iLhBo59iT7W75D7MeregYHtb7O9WyrY9gzWH0B9XItKIWh2OmTn+CHX5yQyS/mS+DH1WxVjWRIvXHpTgL4WrH7zQDga4aMc/UJ0kfh1Sp+T+Ewq23Xi+KQwQNgDuF/Bfo01CbkTed3gxMe6xVwXp1y7hNhvNDbzsIRG5KaH7W20+9HYRcB3ajvWdZyt2W37PN9d9ZNZueZol2/VfZZwwtnCrD3HsjUjL+p7mQjinYl4WsYBz4FLxhWUaW8J17G6SLhZMe4dRFfwn4S9GBcGzS1H/X5HX1uec3D8kOtzEpklfEn9mNKlirEsiRchO81OOv4lUTpyOOfoF6KLxK9T+VyK2JfKdu88v9kLD+GfMiwif2G1g/vyhzw8SeNjnWKuj1OuXUKuQQdjJg4NnC1x264R6ZE9mDosttskju9AR/HVNYg2rvpVwvj2uRdwxlrEokWW75wbmPop1r02/lW2ITy9CeDeJeMmMYOmeI3VJZYbn375b12B9SZg8sD1Q67PSWSW8iXxY5cuqcdyM8ZBjhdE4G5E+HyIj4ToEuvXKX1OGvtS2s43eYDHq4/i50GUMb+o5ROjk9ZEqBEwhjnxsR1ibqhdqhp7rJSSDddjN8fQUV3fi2mnhZI617F2PVcWe1ZLBYtjBYe166nNQh5VzJNxHOuS8SGD61hdJNz49OPamcsVdduC64fcc0pklvIl8WPJ+AkZy80YB4Ahj08uJPLR44n5ivUxSX8Sn0lpu7JrDGRDzuDnB+bDkxkwWcjT/U+srAMlhyQ+1i3mLgjHVIj9Fo0A1CyKwgiRBgPAeolrVop1xpF6nSmpcx37tVUPKZ05xjnWrN8/y8qfxDGzAfpWzZNx6DXL5JSqj9VFwo1PvxAf4XD1AG0b0xfX5yQyS/mS+HGIP0jGcjPGQcjYl3DeLL7KfFHSXxWxL8Z2vgWTAHgLJSx0PZGVu4V6WOgJ6wqoJxsonmYTxpNWx9wZ4ZgKsd+sZPIA95guMdpB6ug3Io2yZM0KYYHON8TxVD3UFe833jA7V6xCSmvVOnbE0I/KzFntXlqpG7hXesc6Bs457nD6PyL4DOHpPLNPl4zQxzkG17G6SLhx+YFP99D6HNSjmlw/5PqcRGYpXxI/pvquYiynjhcujDoC45xlBwnnEr4kfn0zoc9JfCal7cbNzkV6RcAajHkiuzCG9c+setcYlsTHOsVcn2xcu3Db+WIGC3APDBZ6wMKO/B4TON1pnFh8gXX5C41O4/c+NO4Vq795s3NRjqv+rlU3hTMhULQHv08Tx8LK0Vfmw8uVhnBggHzFR2HgkZ+fsT/Q8zZB6jTqSWEJZ2sN5AY4Gi7hU8oTBZeMIVzH6CLhxqdfiNwcrqiXRHH74vqcVGYJX1I/tvuuYiw3YxwYzJIuFcY+pGjv4blOJfLRZvFV5mOS/iQ+k9J2ZS+JAnu+J/5tw0Vuy5Flss+bIj7WJeb6ZOPahdvOFzPYGMAA+hoNuYEXY1uJYSQEFsAsO2Zg64Z+5pSqt+uA4PvY/2JBeepYyD48x7bwGNOXmKqzF6NcwX8qk9jHK/PPR1MOI9Ggt33fvB+dfwv15v5DkvBEwSVjCNcxuki48ekXIjeXK1h4dSSiL67PSWWW8CX1Y6rv1GO5GeMgRy8Gw7do9xORPh+iXxV8lfmYtD+Jz6SwHff11Gtm54r/Bp77EOO8KeJjXWJumWwcu3Db+exfa0AAWEnYH6zU5bz1q6cGuoMMq0aRErttY6weNbmi5j6jG2MpkgNmlfACjIuRx8Ns9Cfz4fnoo2bnc8B1xQ2cCerW0elxQXlVKGoBWOdwXmlQpMJXZjvVuCkM8vmraaEfeCQJ7uUNtgkHkMobU1dQKBQKRd3xfzH3j/yMnyzJAAAFrnRFWHRNYXRoTUwAPG1hdGggeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzE5OTgvTWF0aC9NYXRoTUwiPjxtZnJhYz48bXJvdz48bWk+QzwvbWk+PG1pPm88L21pPjxtaT5uPC9taT48bWk+ajwvbWk+PG1pPnU8L21pPjxtaT5nPC9taT48bWk+YTwvbWk+PG1pPnQ8L21pPjxtaT5lPC9taT48bW8+JiN4QTA7PC9tbz48bWk+bzwvbWk+PG1pPmY8L21pPjxtbz4mI3hBMDs8L21vPjxtaT50PC9taT48bWk+aDwvbWk+PG1pPmU8L21pPjxtbz4mI3hBMDs8L21vPjxtaT50PC9taT48bWk+ZTwvbWk+PG1pPnI8L21pPjxtaT5tPC9taT48bW8+JiN4QTA7PC9tbz48bW8+KDwvbW8+PG1pPm48L21pPjxtaT51PC9taT48bWk+bTwvbWk+PG1pPmU8L21pPjxtaT5yPC9taT48bWk+YTwvbWk+PG1pPnQ8L21pPjxtaT5pPC9taT48bWk+bzwvbWk+PG1pPm48L21pPjxtbz4mI3hBMDs8L21vPjxtaT5vPC9taT48bWk+cjwvbWk+PG1vPiYjeEEwOzwvbW8+PG1pPmQ8L21pPjxtaT5lPC9taT48bWk+bjwvbWk+PG1pPm88L21pPjxtaT5tPC9taT48bWk+aTwvbWk+PG1pPm08L21pPjxtaT5hPC9taT48bWk+dDwvbWk+PG1pPm88L21pPjxtaT5yPC9taT48bW8+KTwvbW8+PG1vPiYjeEEwOzwvbW8+PG1pPnc8L21pPjxtaT5pPC9taT48bWk+dDwvbWk+PG1pPmg8L21pPjxtbz4mI3hBMDs8L21vPjxtaT50PC9taT48bWk+aDwvbWk+PG1pPmU8L21pPjxtbz4mI3hBMDs8L21vPjxtaT5yPC9taT48bWk+bzwvbWk+PG1pPm88L21pPjxtaT50PC9taT48L21yb3c+PG1yb3c+PG1pPkM8L21pPjxtaT5vPC9taT48bWk+bjwvbWk+PG1pPmo8L21pPjxtaT51PC9taT48bWk+ZzwvbWk+PG1pPmE8L21pPjxtaT50PC9taT48bWk+ZTwvbWk+PG1vPiYjeEEwOzwvbW8+PG1pPm88L21pPjxtaT5mPC9taT48bW8+JiN4QTA7PC9tbz48bWk+dDwvbWk+PG1pPmg8L21pPjxtaT5lPC9taT48bW8+JiN4QTA7PC9tbz48bWk+dDwvbWk+PG1pPmU8L21pPjxtaT5yPC9taT48bWk+bTwvbWk+PG1vPiYjeEEwOzwvbW8+PG1vPig8L21vPjxtaT5uPC9taT48bWk+dTwvbWk+PG1pPm08L21pPjxtaT5lPC9taT48bWk+cjwvbWk+PG1pPmE8L21pPjxtaT50PC9taT48bWk+aTwvbWk+PG1pPm88L21pPjxtaT5uPC9taT48bW8+JiN4QTA7PC9tbz48bWk+bzwvbWk+PG1pPnI8L21pPjxtbz4mI3hBMDs8L21vPjxtaT5kPC9taT48bWk+ZTwvbWk+PG1pPm48L21pPjxtaT5vPC9taT48bWk+bTwvbWk+PG1pPmk8L21pPjxtaT5tPC9taT48bWk+YTwvbWk+PG1pPnQ8L21pPjxtaT5vPC9taT48bWk+cjwvbWk+PG1vPik8L21vPjxtbz4mI3hBMDs8L21vPjxtaT53PC9taT48bWk+aTwvbWk+PG1pPnQ8L21pPjxtaT5oPC9taT48bW8+JiN4QTA7PC9tbz48bWk+dDwvbWk+PG1pPmg8L21pPjxtaT5lPC9taT48bW8+JiN4QTA7PC9tbz48bWk+cjwvbWk+PG1pPm88L21pPjxtaT5vPC9taT48bWk+dDwvbWk+PC9tcm93PjwvbWZyYWM+PC9tYXRoPkZMPFcAAAAASUVORK5CYII=)
As with Factoring, this approach will probably lead to being able to cancel a term.
The correct answer is: ![1 half](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABIAAAAjCAYAAACZ6FpfAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAXQ/cXWQAAAJFJREFUeNpjYMAP1IC4FogvMFAIFgNxGhD/Z6ASGDVo1KBBYdB/LHgUDCbwn0w8CgYLsAbiNUD8CYh/QaujaHIMOgjEkUDMA+VrAfFRqBjFQB6IL1HLyz+oYYgl1HsUAQ4gPgmNBLKBIBBvAGI3SgxRghqiQokhGkA8G4i5KDFEHIhXATELpYG7BeoimhZyWAEAI3M1I31CbrEAAABidEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1mcmFjPjxtbj4xPC9tbj48bW4+MjwvbW4+PC9tZnJhYz48L21hdGg+ND6jrQAAAABJRU5ErkJggg==)
We first try substitution:
=
= ![0 over 0](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABIAAAAjCAYAAACZ6FpfAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAXQ/cXWQAAAJtJREFUeNpjYMAN+IB4GhCfhOKZUDGSwV4g9kPi+0DFSAKhQDwJi/gEII4kxaA1QOyGRdwRKkc0eAfEbFjEQWIvSTHoDx65X9Qy6AcpBr3E4TUOUr0GClAPLOJupAZ2EBDPxiI+n9ToB4H9QFwAxCxQb1YD8UFyUrYgEM+FBu43aBbhYRgFgwv8JxOPgsEERquj0epotDqiXnUEAHl3QY/fxbExAAAAYnRFWHRNYXRoTUwAPG1hdGggeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzE5OTgvTWF0aC9NYXRoTUwiPjxtZnJhYz48bW4+MDwvbW4+PG1uPjA8L21uPjwvbWZyYWM+PC9tYXRoPt4dSgEAAAAASUVORK5CYII=)
Since the limit is in the form
, it is indeterminate—we don’t yet know what is it. We need to do some work to put it in a form where we can determine the limit.
So let’s get rid of the square roots, using the conjugate just like you practiced in algebra: multiply both the numerator and denominator by the conjugate of the numerator ![square root of x plus 1 space end root plus 1](data:image/png;base64,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)
.
![L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator square root of x plus 1 end root minus 1 over denominator x end fraction](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIkAAAAnCAYAAAA7MmtXAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAbSkFbcgAAAvlJREFUeNrtm09o1EAUxodSisgelCoetCgiRUR6UqqoeOmhSA9SetCjKCJFZG/1oCj0oCcV9dZKKaUK4j9EVBApIiKIqHiQpVI8iQdBQaRCUdfvkbcQQ5JJNtlkZ+b94GOTvGRnMpnMvHmZUcpdKlDdIAklcACak2IQ4ngEHZFiEKJYBX3nX0EI5Rh0t43y0wudht5ZnqZRkC8y0kb5mYGOFuyglpGmMWyAvkLLWphGveDrysir1ZyEJmLsh6HzIcfH2SaVxAHeQoOac95Aa3z7h6Cr0pLYRyXk2FboC9SpuZYq0QXeHoCeSndjF5ugKb753QHbOehiwv95Au1l7787QUHnETGtp3y4RadpDae4klAc5FXA9gnalvB/qtAS1CeOq72MQn99vsUO6GOCrobYBd3iLuegVBK7WYSu8/YV6GyCazZC96Hl7Ne8TtDdSCUxmEnoF7ceFBvZrDl/NfQwUCmGlBd4kkoSePs6LKkkK6A/0G3opebcLugOtDbEdiPBsDnrgyp6akDTadLI4L3G/sKwivKcC6CqhFzYz29NFMPQrGH3tJ0ryTp5vPlwBhqLsfmbpjGD7mtEHm1+3OTWpFm74ADzEY5bgxq0XorJXWhE81NjX2zj/NdFrZ983a/iJwaT/Zm8S25D4ecpjX26xPwNcXdX422hBC4rb95EFJeg4yXlbSe3Yt2sOT4mFAxFG+OiijTp5lqMnYJWfT7/5XEKJ5dasEqM/Z7yPts32KO87ytCwXyLcWAoZL0F+uzbD+sOGoE4+no6nCLtCR5Z9UfYf6j/PxV08DHBQGiCTlVFzyEd1XjTv5U31T9s5BJkSYrbXL+mFtHS6FqSBeWF0JO0JJ3SkpjJAHc3H5T3yT0NaX0S2n4gRW4WPTz6qHCXMt6C0Q2NaFZyBaS09hlUPnks0zCaLh7Z9Pr2FzQtQ7Nxknl2sE8YWE5Zl2kIDpB1mYbgCGmWaQiOkmWZhuAAWZdpCJaTxzINwWLyWqYhWBweKGuZRm78A/5ZSFM4QSsYAAABEXRFWHRNYXRoTUwAPG1hdGggeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzE5OTgvTWF0aC9NYXRoTUwiPjxtaT5MPC9taT48bXN1Yj48bWk+dDwvbWk+PG1yb3c+PG1pPng8L21pPjxtbyBzdHJldGNoeT0iZmFsc2UiPiYjeDIxOTI7PC9tbz48bW4+MDwvbW4+PC9tcm93PjwvbXN1Yj48bWZyYWM+PG1yb3c+PG1zcXJ0PjxtaT54PC9taT48bW8+KzwvbW8+PG1uPjE8L21uPjwvbXNxcnQ+PG1vPiYjeDIyMTI7PC9tbz48bW4+MTwvbW4+PC9tcm93PjxtaT54PC9taT48L21mcmFjPjwvbWF0aD5TaL2gAAAAAElFTkSuQmCC)
x ![fraction numerator square root of x plus 1 space end root space plus 1 over denominator square root of x plus 1 space end root space plus 1 end fraction](data:image/png;base64,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)
![L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator left parenthesis square root of x plus 1 end root space right parenthesis squared minus 1 over denominator x left parenthesis square root of x plus 1 end root space plus 1 right parenthesis end fraction](data:image/png;base64,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)
![L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator x over denominator x left parenthesis square root of x plus 1 end root space plus 1 right parenthesis end fraction](data:image/png;base64,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)
![L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator 1 over denominator left parenthesis square root of x plus 1 end root space plus 1 right parenthesis end fraction](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJcAAAAqCAYAAAC+y8nSAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAXQ/cXWQAAA25JREFUeNrtm09o1EAUxociRWQPShUPKopIEZFS0FJFi5ceRHoQ6cEeRSkipeytHhSFHiw9VNHeVIoULYj/EFFBpMhSCiIqHmRRiifxICgUqVCq63vdJ8SYTGayzSbpfD/42CRvNzs7eftm3ptEKZAVmklnSW/RFWCpGSf1kiroCpAUcC4A5wJwLgDgXADOBeBcAMC5AJwLwLkyzhypAdc4Fafya1mxjfQuwj4FPwBxOEya0NiPkG6im0AczpEGNDZvyB5AdwEbbkv0imsHIJQPpA0ae5m0Gd0EbOEM8UeEfS5n2RaUjv6jnTSpuXhsf4H/IIhDD2kswn4jxfZ1ybBclm2QIy6Tjmnsl0h9KbVtr0TNJtGkHAM54R7poMY+SrqusZdILZ752VOLyT9HzILG/oB0wLPfQXqIS5YfvmkmaI2kHaTPnv2gYetvAXZEVQuuplyVTLU9xD6r/l2SapBjwCH4saiiOEsQpyKyjAVVfcQqKBP0M59A+wt5z8qW+7ytHBLZoiLXDKnNMHKtSChyHY3ImEFKdMqw+J60zvKztnMu3n5kcF6OlEMW7XhCOoFLacSQ9G/ibJJsriBfOJhAtsgRZY04Ln/XoYjPcHIxbfEda0nf5RWYMeVJ4hKhUTLFZs/+TEQkikOXTPo58eg3/OGtFuc/SbqfoQuXhcf7o9qwS669U7Qq+/vOODJ2Z+g3ZOHxfpM2lMTJnGFY2RV8t5C+klYm2KZKnT9Xrzb0W85rcw8XcDss3n9aUz5hjod04KDYXHYung8/d8m5Zi3LIW+UfnWCeU1a79nnpbJRRK7FfnaqoL0QcjwoydhJ+qKqtTMd7Hwjst0Z49+6XJ2LmXfZufjBkjHppP0+2wXSRcPzPlPVGhtnT00GF2QpKuyVGp2pHm1wyrn8w+IZcS6uY730vfcTabfheYvSkS0ZihoYFusMR5h9Ace5wPvbM3faQ/poMCQqOd8dGRp74FzuTuh1pQi+VfuWbF8hnTc431ZVvcVnlczbXhkMi644V5/0tzPoiqjXSD8lWnFta3vEuXi56bHPmXi1YBzOtYhzRVRmOmRutJr0i3RXRa898nyCb6IMeiJqwqB8UasjpH2LTVQbnFz+UUq/cF2SjioqUAuJL1xnGV6MDqqst4lzbYR/1DSv7UU3BNONLkiWP818fkw/KafvAAABLXRFWHRNYXRoTUwAPG1hdGggeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzE5OTgvTWF0aC9NYXRoTUwiPjxtaT5MPC9taT48bXN1Yj48bWk+dDwvbWk+PG1yb3c+PG1pPng8L21pPjxtbyBzdHJldGNoeT0iZmFsc2UiPiYjeDIxOTI7PC9tbz48bW4+MDwvbW4+PC9tcm93PjwvbXN1Yj48bWZyYWM+PG1uPjE8L21uPjxtcm93Pjxtbz4oPC9tbz48bXNxcnQ+PG1pPng8L21pPjxtbz4rPC9tbz48bW4+MTwvbW4+PC9tc3FydD48bW8+JiN4QTA7PC9tbz48bW8+KzwvbW8+PG1uPjE8L21uPjxtbz4pPC9tbz48L21yb3c+PC9tZnJhYz48L21hdGg+RQUaewAAAABJRU5ErkJggg==)
Put the value x = 0 so,
= ![1 half](data:image/png;base64,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)
Therefore limit value of
= ![1 half](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABIAAAAjCAYAAACZ6FpfAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAXQ/cXWQAAAJFJREFUeNpjYMAP1IC4FogvMFAIFgNxGhD/Z6ASGDVo1KBBYdB/LHgUDCbwn0w8CgYLsAbiNUD8CYh/QaujaHIMOgjEkUDMA+VrAfFRqBjFQB6IL1HLyz+oYYgl1HsUAQ4gPgmNBLKBIBBvAGI3SgxRghqiQokhGkA8G4i5KDFEHIhXATELpYG7BeoimhZyWAEAI3M1I31CbrEAAABidEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1mcmFjPjxtbj4xPC9tbj48bW4+MjwvbW4+PC9tZnJhYz48L21hdGg+ND6jrQAAAABJRU5ErkJggg==)
Related Questions to study
If
then
If
then
We can only apply the L’Hospital’s rule if the direct substitution returns an indeterminate form, that means or
.
We can only apply the L’Hospital’s rule if the direct substitution returns an indeterminate form, that means or
.
Hence Choice 4 is correct
Hence Choice 4 is correct
We can only apply the L’Hospital’s rule if the direct substitution returns an indeterminate form, that means or
.
We can only apply the L’Hospital’s rule if the direct substitution returns an indeterminate form, that means or
.
We can only apply the L’Hospital’s rule if the direct substitution returns an indeterminate form, that means .
We can only apply the L’Hospital’s rule if the direct substitution returns an indeterminate form, that means .