Chapter 2 - Limits - 2.4 Infinite Limits - 2.4 Exercises - Page 87: 30

$a-\\\lim\limits_{x \to 7^-}\frac{x+7}{x^4-49x^2}=\lim\limits_{x \to 7^-}\frac{x+7}{x^2(x+7)(x-7)}=\lim\limits_{x \to 7^-}\frac{1}{x^2(x-7)}=-\infty \\ (the\,denominator\,is\,negative\,and\,approach\,\,\,0)\\ (so\,\,there\,is\,\,vertical\,asymptote\,at\,x=7 )\\ b-\\\lim\limits_{x \to 7^+}\frac{x+7}{x^4-49x^2}=\lim\limits_{x \to 7^+}\frac{x+7}{x^2(x+7)(x-7)}=\lim\limits_{x \to 7^+}\frac{1}{x^2(x-7)}=\infty \\ (the\,denominator\,is\,positive\,and\,approach\,\,\,0)\\ (so\,\,again\,there\,is\,\,vertical\,asymptote\,at\,x=7 )\\ c-\\\lim\limits_{x \to -7}\frac{x+7}{x^4-49x^2}=\lim\limits_{x \to -7}\frac{x+7}{x^2(x+7)(x-7)}=\lim\limits_{x \to -7}\frac{1}{x^2(x-7)}=-\frac{1}{686} \\ since\,the\,limit\,exists\,\\ there\,is\,\,no\,\,vertical\,asymptote\,at\,x=-7 $ $d-\\ \lim\limits_{x \to 0}\frac{x+7}{x^4-49x^2}=\lim\limits_{x \to 0}\frac{x+7}{x^2(x+7)(x-7)}=\lim\limits_{x \to 0}\frac{1}{x^2(x-7)}=-\infty \\ (the\,denominator\,is\,negative\,and\,approach\,\,\,0)\\ (so\,\,there\,is\,\,vertical\,asymptote\,at\,x=0 )\\ $

$a-\\\lim\limits_{x \to 7^-}\frac{x+7}{x^4-49x^2}=\lim\limits_{x \to 7^-}\frac{x+7}{x^2(x+7)(x-7)}=\lim\limits_{x \to 7^-}\frac{1}{x^2(x-7)}=-\infty \\ (the\,denominator\,is\,negative\,and\,approach\,\,\,0)\\ (so\,\,there\,is\,\,vertical\,asymptote\,at\,x=7 )\\ b-\\\lim\limits_{x \to 7^+}\frac{x+7}{x^4-49x^2}=\lim\limits_{x \to 7^+}\frac{x+7}{x^2(x+7)(x-7)}=\lim\limits_{x \to 7^+}\frac{1}{x^2(x-7)}=\infty \\ (the\,denominator\,is\,positive\,and\,approach\,\,\,0)\\ (so\,\,again\,there\,is\,\,vertical\,asymptote\,at\,x=7 )\\ c-\\\lim\limits_{x \to -7}\frac{x+7}{x^4-49x^2}=\lim\limits_{x \to -7}\frac{x+7}{x^2(x+7)(x-7)}=\lim\limits_{x \to -7}\frac{1}{x^2(x-7)}=-\frac{1}{686} \\ since\,the\,limit\,exists\,\\ there\,is\,\,no\,\,vertical\,asymptote\,at\,x=-7 $ $d-\\ \lim\limits_{x \to 0}\frac{x+7}{x^4-49x^2}=\lim\limits_{x \to 0}\frac{x+7}{x^2(x+7)(x-7)}=\lim\limits_{x \to 0}\frac{1}{x^2(x-7)}=-\infty \\ (the\,denominator\,is\,negative\,and\,approach\,\,\,0)\\ (so\,\,there\,is\,\,vertical\,asymptote\,at\,x=0 )\\ $