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Homework Statement
Prove that the function:
[tex]\frac{2x-1}{x^2+1}, x \in \mathbb{R}[/tex]
is continuous.
Homework Equations
Definition 1.
The function y=f(x) satisfied by the set Df is continuous in the point x=a only if:
10 f(x) is defined in the point x=a i.e. [itex]a \in D_f[/itex]
20 there is bound [tex]\lim_{x \rightarrow a}f(x)[/tex]
30 [tex]\lim_{x \rightarrow a}f(x)=f(a)[/tex]
Theorem 1.
If the functions y=f(x) and y=g(x) are continuous in the point x=a Є Df ∩ Dg, then in the point x=a these functions are continuous:
y=f(x)+g(x), y=f(x)g(x) and y=f(x)/g(x), if g(a) ≠ 0.
The Attempt at a Solution
I tried using the definition 1.
But also this function is composition of two functions f(x) and g(x), so can I use the fact that f(x)=2x-1 and g(x)=x2+1 are continuous, and y=f(x)/g(x), g(a) ≠ 0 since x2+1 ≠ 0 ?