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What is the limit of $\sin^{-1} (\sec x) $ as $x$ tends to $0$.

By direct substitution the value of $\sec x $ at $x =0$ is $1$. So limit should be $1$.

But answer is given that limit doesn't exist. How $?$

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First of all, if you do direct substitution, then yes, $\sec 0 = 1,$ but you still have to consider the arc sine function. And $\sin^{-1} (1) = \frac\pi2.$ So the result of direct substition by $x=0$ is $\sin^{-1} (\sec x) = \frac\pi2.$

But the direct substitution method agrees with the limit only if both of the following conditions are met:

  • The limit exists.
  • The function is continuous.

The existence of the limit as $x\to0$ doesn't depend on the value at $x=0.$ But it does depend on the value of the function when $x$ is very close to zero. And if $x$ is close to zero, but not equal to zero, we have

$$ \sec x > 1 $$

and since $\sin$ can only produce numbers in the range $[-1,1],$ $\sin^{-1} (\sec x)$ is undefined when $\sec x > 1.$

You can't produce a limit when the function is not defined anywhere near the limit point except at the limit point itself.

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Direct substitution does not always work. It only works when the given function is continuous about $x_0$ (the number x approaches, in this case 0). Remember the idea of limits is that when x gets "close" to $0, f(x)$ ought to get "close" to some number. In the case of this problem, $\sin^{-1}(x)$ is only defined on the domain $[-1, 1]$, but the range of $sec(x)$ is $(-\infty, -1] \cup [1, \infty)$. Thus, the domain of $\sin^{-1}(\sec(x))$ is only a set of discrete points $\{n\pi : n \in \mathbb{Z}\}$. If you try plotting $\sin^{-1}(\sec(x))$ in desmos, maybe this will be more clear. With just these single points, it is impossible to say what the behavior of $\sin^{-1}(\sec(x))$ is like "close to" $x=0$ as the function is not defined close to $x = 0$. Hence why the limit does not exist.

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  • $\begingroup$ I'd say the domain of $\sin^{-1} (\sec x) $ is the set $\{n\pi\mid n\in\mathbb N\},$ whose image under $\sec$ is $\{-1,1\}.$ $\endgroup$ – David K Jul 22 at 2:33
  • $\begingroup$ Thank you that's more accurate. $\endgroup$ – zjm Jul 22 at 2:41

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