lim _(xarrow 0)(1-5x)^(2)/(3/n)x=

To evaluate the limit \(\lim_{x \to 0} (1 - 5x)^{\frac{2}{3/n}x}\), we can use the exponential function and properties of limits.<br /><br />First, let's rewrite the expression in a more convenient form. We have:<br /><br />\[<br />(1 - 5x)^{\frac{2}{3/n}x}<br />\]<br /><br />We can express this using the exponential function \(e\):<br /><br />\[<br />(1 - 5x)^{\frac{2}{3/n}x} = e^{\frac{2}{3/n} \ln(1 - 5x)}<br />\]<br /><br />Now, we need to evaluate the limit of the exponent:<br /><br />\[<br />\lim_{x \to 0} \frac{2}{3/n} \ln(1 - 5x)<br />\]<br /><br />As \(x \to 0\), \(1 - 5x \to 1\), and \(\ln(1 - 5x) \to \ln(1) = 0\). Therefore, we have:<br /><br />\[<br />\lim_{x \to 0} \frac{2}{3/n} \ln(1 - 5x) = \frac{2}{3/n} \cdot 0 = 0<br />\]<br /><br />So, the expression simplifies to:<br /><br />\[<br />e^0 = 1<br />\]<br /><br />Therefore, the limit is:<br /><br />\[<br />\lim_{x \to 0} (1 - 5x)^{\frac{2}{3/n}x} = 1<br />\]