770. 1) lim _(xarrow 0)(arctgx)/(x) y k a 3 a HHe. Ilo.JozKHT (1-2x)-alpha

To find the limit of the, we can use L'Hopital's rule. L'Hopital's rule states that if the limit of a function is in the form of 0/0 or ∞/∞, we can differentiate the numerator and denominator separately and then take the limit again.<br /><br />In this case, the given expression is $\lim _{x\rightarrow 0}\frac {arctgx}{x}$. As x approaches 0, both the numerator and denominator approach 0, so we can apply L'Hopital's rule.<br /><br />Differentiating the numerator with respect to x, we get $\frac{1}{1+x^2}$. Differentiating the denominator with respect to x, we get 1.<br /><br />Now, we can take the limit of the expression $\frac{\frac{1}{1+x^2}}{1}$ as x approaches 0.<br /><br />Substituting x = 0 into the expression, we get $\frac{1}{1+0^2} = 1$.<br /><br />Therefore, the limit of the given expression is 1.