evaluate the following limit. write out an explanation as to how you got your answer. be sure to justify…

evaluate the following limit. write out an explanation as to how you got your answer. be sure to justify your work by stating any theorems used.\n\\(\\lim_{z\\to0}4z\\sin(\\frac{1}{z^{2}})=\\)

evaluate the following limit. write out an explanation as to how you got your answer. be sure to justify your work by stating any theorems used.\n\\(\\lim_{z\\to0}4z\\sin(\\frac{1}{z^{2}})=\\)

Answer

Explanation:

Step1: Recall the Squeeze - Theorem

The Squeeze - Theorem states that if (g(x)\leq f(x)\leq h(x)) for all (x) in some open interval containing (a) (except possibly at (x = a)) and (\lim_{x\rightarrow a}g(x)=\lim_{x\rightarrow a}h(x)=L), then (\lim_{x\rightarrow a}f(x)=L). We know that (- 1\leq\sin\left(\frac{1}{x^{2}}\right)\leq1) for all (x\neq0).

Step2: Multiply the inequality by (4x)

Multiply the inequality (-1\leq\sin\left(\frac{1}{x^{2}}\right)\leq1) by (4x). We get (-4x\leq4x\sin\left(\frac{1}{x^{2}}\right)\leq4x) for (x > 0) and (4x\leq4x\sin\left(\frac{1}{x^{2}}\right)\leq - 4x) for (x<0).

Step3: Find the limits of the bounding functions

We find (\lim_{x\rightarrow0}(-4x)=0) and (\lim_{x\rightarrow0}(4x)=0). By the Squeeze - Theorem, since (\lim_{x\rightarrow0}(-4x)=\lim_{x\rightarrow0}(4x) = 0) and (-4x\leq4x\sin\left(\frac{1}{x^{2}}\right)\leq4x) (or (4x\leq4x\sin\left(\frac{1}{x^{2}}\right)\leq - 4x) for (x < 0)), we have (\lim_{x\rightarrow0}4x\sin\left(\frac{1}{x^{2}}\right)=0).

Answer:

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