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Show that $ f $ is continuous on $ (-\infty, \infty ) $.

$ f(x) = \left\{ \begin{array}{ll} 1 - x^2 & \mbox{if $ x \le 1 $}\\ \ln x & \mbox{if $ x > 1 $} \end{array} \right.$

$f(x)=\left\{\begin{array}{ll}1-x^{2} & \text { if } x \leq 1 \\ \ln x & \text { if } x>1\end{array}\right.$By Theorem $5,$ since $f(x)$ equals the polynomial $1-x^{2}$ on $(-\infty, 1], f$ is continuous on $(-\infty, 1]$By Theorem $7,$ since $f(x)$ equals the logarithm function $\ln x$ on $(1, \infty), f$ is continuous on $(1, \infty)$At $x=1, \lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1^{-}}\left(1-x^{2}\right)=1-1^{2}=0$ and $\lim _{x \rightarrow 1^{+}} f(x)=\lim _{x \rightarrow 1^{+}} \ln x=\ln 1=0 .$ Thus, $\lim _{x \rightarrow 1} f(x)$ exists andequals $0 .$ Also, $f(1)=1-1^{2}=0 .$ Thus, $f$ is continuous at $x=1 .$ We conclude that $f$ is continuous on $(-\infty, \infty)$

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 5

Continuity

Limits

Derivatives

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This is problem number thirty nine of this to a Calculus East addition. Section two point five show that if his continuous on the interval from negative infinity to Infinity and the function F is this piece for its function one minus X squared. If X is less than or equal to one, Ellen of X X is greater than one. Now we recall the definition of continuity that states, if a function have is continuous at a point. A. The limit is, experts say the function F is equal to have evaluated at A. And it's important to also know that this is, um, true if and only if f is continuous at a. Which means that if we find this statement teacher, then we can also make a statement that if it's continuous at that certain point, which is something that we are attempting to do in this problem. So we know that dysfunction one minus X squared is continuous, um, on its domain here from X Brexit's listener equal to one financial. Augie's also continuous on its domain from X for X is greater than one. That's all a matter of what's happening in that X equals to one, which is our A. So we have one function on the list of conviction on the right. We want to know Lim as X approaches one from the letters and the limit is expressions Warner from the rain. We know that from the left and repeating the one minus X great function and using our limit loans, we are able to reduce this town too. Ah, the limit is expecting one. A purchase one from the left of one, which is one rightness on the the minutes experts One on the left of banks which is negative one right, which is one squared, which is what because is zero now for the Limited Express is one of the rink Mary's in the natural or function and we never characteristically using our understanding the function natural log. And it's Graf And as we approach one from the right that the natural our function approaches value zero. We see that from the left on the right, this limit perches zero. So we stayed limit as experts is one of this function. F is equal to zero. And this is precisely function. And, you know, I waited at zero. No, sorry, is valued it at one and that this is true as we stayed before, then it means and it confirms the dysfunction is definitely continuous. Okay, Point X equals one. And we already discussed that this function is continuous for anybody lesson equal to one and greater than one. And so we have shown in this case that this function is a continuous on the domain of negative infinity to infinity.

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