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

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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