if f(x) = \( \{mx^2+n,x<0,nx+m,0≤x≤1,nx^3+m,x>1\)

For what integers m and n does \( \lim\limits_{x \to 0}f(x)\) and\( \lim\limits_{x \to 1}f(x)\) exists?

Asked by Pragya Singh | 1 year ago |  82

1 Answer

Solution :-

\( \lim\limits_{x \to 0}f(x)=\lim\limits_{x \to 0}(mx^2+n)\)

\( m(0)^2+n=n\) 

\( \lim\limits_{x \to 0^+}f(x)=\lim\limits_{x \to 0}(nx+m)\)

\( n(0)+m=m\)

Thus,\( \lim\limits_{x \to 0^+}f(x)\) exists if m=n

\( \lim\limits_{x \to 1^-}f(x)=\lim\limits_{x \to 1}(nx+m)\)

= n(1)+m

= m+n

\( \lim\limits_{x \to 1^+}f(x)=\lim\limits_{x \to 1}(nx^3+m)\)

= n(1)3+m

= m+n

\( \lim\limits_{x \to 1^-}f(x)= \lim\limits_{x \to 1^+}f(x)= \lim\limits_{x \to 1}f(x)\)

Thus,\( \lim\limits_{x \to 1}f(x)\) exists for any internal value of m and n.

Answered by Abhisek | 1 year ago

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