I have to verify the continuity of this function and I don't really understand anything...
Here it is (sorry but I don't know the specific text formatting so it's gonna be raw): $$ f(x)= \begin{cases} 1 + 2^{1 / (x-1)} & x < 1 \\ 1 & x \geq 1 \end{cases} $$ Verify continuity for $a = 1$.
Also this: $$ f(x) = \begin{cases} e^{-(1 / (x+1)^2)} & x \neq 2 \\ 0 & x = 2 \end{cases} $$
Verify continuity for $a = 2$.
Please explain as well as you can. How is the first one not $1+2^\infty= \infty$ ? The answers at the end of the book is $0$ I think.
A function is continuous if the left limit is equal to the right limit and it is equal to the function value evaluated at the point.
$$f(1) = 1$$ $$\lim_{x \rightarrow 1^+} f(x) = \lim_{x \rightarrow 1^+} 1 = 1$$
$$\lim_{x \rightarrow 1^-} f(x) = \lim_{x \rightarrow 1^+} 1+2^{\frac1{x-1}} = 1+2^{\lim_{x \rightarrow 1^-}\frac1{x-1}}=1+2^{-\infty}=1$$
Hence it is continuous.
For the second function.
$$f(2)=0$$ $$\lim_{x \rightarrow 2}\exp\left(-\frac1{(x+1)^2}\right)=\exp\left( -\frac19\right) \neq 0$$
Hence it is not continuous.