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**redsoxfan325** For the first question, is it as $\displaystyle x\to 0$?

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For the second question, let $\displaystyle \{x_n\}$ be a sequence that converges to $\displaystyle x$ and $\displaystyle x'$. (We want to show that $\displaystyle x=x'$.) Because of convergence, $\displaystyle \forall~\epsilon>0, \exists~ N$ such that $\displaystyle n>N \implies d(x,x_n)<\frac{\epsilon}{2}$, and $\displaystyle \exists~ N'$ such that $\displaystyle n>N' \implies d(x_n,x')<\frac{\epsilon}{2}$.

So if $\displaystyle n>\max\{N,N'\}$, by the triangle inequality, $\displaystyle d(x,x')\leq d(x,x_n)+d(x_n,x')<\frac{\epsilon}{2}+\frac{\epsil on}{2} = \epsilon$. Since $\displaystyle \epsilon$ was arbitrary, we can conclude that $\displaystyle x=x'$.

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For the third question, you know this function is continuous at $\displaystyle x=1$ (because it is a composition of continuous functions), so you can just plug in $\displaystyle 1$ for $\displaystyle x$ to get $\displaystyle \sec( 1\cdot\sec^2(1) - \tan^2(1) - 1)=\sec(0)=1$

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For the fourth question, $\displaystyle f(-4) = -3$ and $\displaystyle f(-3) = 19$, so by the Intermediate Value Theorem $\displaystyle \exists~x_1\in(-4,-3)$ such that $\displaystyle f(x_1)=0$.

Similarly, $\displaystyle f(1)=-13$ so the IMT says that $\displaystyle \exists~x_2\in(-3,1)$ such that $\displaystyle f(x_2)=0$.

Similarly, $\displaystyle f(4)=5$ so the IMT says that $\displaystyle \exists~x_3\in(1,4)$ such that $\displaystyle f(x_3)=0$.

Thus there are 3 zeroes in $\displaystyle (-4,4)$.

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I hope I was some help.