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Let it be known that

lim_(x rarr1)f(x)=+oo,lim_(x rarr1)g(x)=-oo,lim_(x rarr1)v(x)=0", "
Find the values of the following limits:
3) 
lim_(x rarr1)(g(x)+v(x));
4) 
lim_(x rarr1)(1)/(g(x));
5) 
lim_(x rarr1)(v(x))/(f(x));

Let it be known that\newlinelimx1f(x)=+,limx1g(x)=,limx1v(x)=0 \lim _{x \rightarrow 1} f(x)=+\infty, \lim _{x \rightarrow 1} g(x)=-\infty, \lim _{x \rightarrow 1} v(x)=0 \text {, } \newlineFind the values of the following limits:\newline33) limx1(g(x)+v(x)) \lim _{x \rightarrow 1}(g(x)+v(x)) ;\newline44) limx11g(x) \lim _{x \rightarrow 1} \frac{1}{g(x)} ;\newline55) limx1v(x)f(x) \lim _{x \rightarrow 1} \frac{v(x)}{f(x)} ;

Full solution

Q. Let it be known that\newlinelimx1f(x)=+,limx1g(x)=,limx1v(x)=0 \lim _{x \rightarrow 1} f(x)=+\infty, \lim _{x \rightarrow 1} g(x)=-\infty, \lim _{x \rightarrow 1} v(x)=0 \text {, } \newlineFind the values of the following limits:\newline33) limx1(g(x)+v(x)) \lim _{x \rightarrow 1}(g(x)+v(x)) ;\newline44) limx11g(x) \lim _{x \rightarrow 1} \frac{1}{g(x)} ;\newline55) limx1v(x)f(x) \lim _{x \rightarrow 1} \frac{v(x)}{f(x)} ;
  1. Limit Sum Property: limx1(g(x)+v(x))\lim_{x \to 1}(g(x)+v(x)) is the sum of the limits of g(x)g(x) and v(x)v(x) as xx approaches 11.\newlinelimx1g(x)=\lim_{x \to 1}g(x) = -\infty and limx1v(x)=0\lim_{x \to 1}v(x) = 0.\newlineSo, limx1(g(x)+v(x))=+0=\lim_{x \to 1}(g(x)+v(x)) = -\infty + 0 = -\infty.
  2. Reciprocal Limit Property: limx11g(x)\lim_{x \rightarrow 1}\frac{1}{g(x)} is the reciprocal of the limit of g(x)g(x) as xx approaches 11.\lim_{x \rightarrow 1}g(x) = -\infty\.So, limx11g(x)=1=0\lim_{x \rightarrow 1}\frac{1}{g(x)} = \frac{1}{-\infty} = 0.
  3. Ratio Limit Property: limx1v(x)f(x)\lim_{x \to 1}\frac{v(x)}{f(x)} is the ratio of the limits of v(x)v(x) and f(x)f(x) as xx approaches 11.\newlinelimx1v(x)=0\lim_{x \to 1}v(x) = 0 and limx1f(x)=+\lim_{x \to 1}f(x) = +\infty.\newlineSo, limx1v(x)f(x)=0+=0\lim_{x \to 1}\frac{v(x)}{f(x)} = \frac{0}{+\infty} = 0.

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