Chapter 1 - Limits and Their Properties - 1.3 Exercises - Page 69: 112

(a) Let a function $g(x)=|x|$ Now let $\lim_\limits{x\to c}|f(x)|=0$ Using g function. We get, $\lim_\limits{x\to c}g(f(x))=0$ Now by the property of limit. We get, $g\left(\lim_\limits{x\to c}f(x)\right)=0$ We know that a modulus function is zero only at the origin. Hence, $\lim_\limits{x\to c}f(x)=0$ (b) Again let a function $g(x)=|x|$ And also, let $\lim_\limits{x\to c}f(x)=L$ We need to prove $\lim_\limits{x\to c}|f(x)|=|L|$ Which can also be written as $\lim_\limits{x\to c}g(f(x))=g(L)$ Now use the property of limit on the right-hand side. We get, $\lim_\limits{x\to c}g(f(x))=g\left(\lim_\limits{x\to c}f(x)\right)$ $\implies \lim_\limits{x\to c}g(f(x))=g(L)=L.H.S$ Hence proved

(a) Let a function $g(x)=|x|$ Now let $\lim_\limits{x\to c}|f(x)|=0$ Using g function. We get, $\lim_\limits{x\to c}g(f(x))=0$ Now by the property of limit. We get, $g\left(\lim_\limits{x\to c}f(x)\right)=0$ We know that a modulus function is zero only at the origin. Hence, $\lim_\limits{x\to c}f(x)=0$ (b) Again let a function $g(x)=|x|$ And also, let $\lim_\limits{x\to c}f(x)=L$ We need to prove $\lim_\limits{x\to c}|f(x)|=|L|$ Which can also be written as $\lim_\limits{x\to c}g(f(x))=g(L)$ Now use the property of limit on the right-hand side. We get, $\lim_\limits{x\to c}g(f(x))=g\left(\lim_\limits{x\to c}f(x)\right)$ $\implies \lim_\limits{x\to c}g(f(x))=g(L)=L.H.S$ Hence proved