If lim _(xarrow a)(a^3-5a^2x)/(a^2)-25x^(2)=lim _(xarrow 1)(x^4-1)/(x^3)-1 then find a. Rationalization

To solve this problem, we need to find the value of \( a \) such that the two limits are equal.<br /><br />Given:<br />\[ \lim_{x \to a} \frac{a^3 - 5a^2x}{a^2 - 25x^2} = \lim_{x \to 1} \frac{x^4 - 1}{x^3 - 1} \]<br /><br />First, let's evaluate the right-hand side limit:<br />\[ \lim_{x \to 1} \frac{x^4 - 1}{x^3 - 1} \]<br /><br />We can factorize the numerator and denominator:<br />\[ x^4 - 1 = (x^2 - 1)(x^2 + 1) = (x - 1)(x + 1)(x^2 + 1) \]<br />\[ x^3 - 1 = (x - 1)(x^2 + x + 1) \]<br /><br />Thus,<br />\[ \frac{x^4 - 1}{x^3 - 1} = \frac{(x - 1)(x + 1)(x^2 + 1)}{(x - 1)(x^2 + x + 1)} \]<br /><br />Cancel the common factor \((x - 1)\):<br />\[ \frac{(x + 1)(x^2 + 1)}{x^2 + x + 1} \]<br /><br />Now, evaluate the limit as \( x \to 1 \):<br />\[ \lim_{x \to 1} \frac{(x + 1)(x^2 + 1)}{x^2 + x + 1} = \frac{(1 + 1)(1^2 + 1)}{1^2 + 1 + 1} = \frac{2 \cdot 2}{1 + 1 + 1} = \frac{4}{3} \]<br /><br />So, we have:<br />\[ \lim_{x \to a} \frac{a^3 - 5a^2x}{a^2 - 25x^2} = \frac{4}{3} \]<br /><br />Next, let's evaluate the left-hand side limit:<br />\[ \lim_{x \to a} \frac{a^3 - 5a^2x}{a^2 - 25x^2} \]<br /><br />Factorize the numerator and denominator:<br />\[ a^3 - 5a^2x = a^2(a - 5x) \]<br />\[ a^2 - 25x^2 = a^2(1 - 25x^2/a^2) = a^2(1 - (5x/\sqrt{a^2}))^2 \]<br /><br />Thus,<br />\[ \frac{a^3 - 5a^2x}{a^2 - 25x^2} = \frac{a^2(a - 5x)}{a^2(1 - (5x/\sqrt{a^2}))^2} = \frac{a - 5x}{(1 - (5x/\sqrt{a^2}))^2} \]<br /><br />Now, evaluate the limit as \( x \to a \):<br />\[ \lim_{x \to a} \frac{a - 5x}{(1 - (5x/\sqrt{a^2}))^2} = \frac{a - 5a}{(1 - (5a/\sqrt{a^2}))^2} = \frac{-4a}{(1 - 5/\sqrt{a})^2} \]<br /><br />Set this equal to \(\frac{4}{3}\):<br />\[ \frac{-4a}{(1 - 5/\sqrt{a})^2} = \frac{4}{3} \]<br /><br />Solve for \( a \):<br />\[ -4a = \frac{4(1 - 5/\sqrt{a})^2}{3} \]<br />\[ -3a = (1 - 5/\sqrt{a})^2 \]<br />\[ 3a = (5/\sqrt{a} - 1)^2 \]<br />\[ \sqrt{a} = \frac{5 - \sqrt{3a}}{3} \]<br /><br />Square both sides:<br />\[ a = \left(\frac{5 - \sqrt{3a}}{3}\right)^2 \]<br /><br />This is a quadratic equation in \( \sqrt{a} \). Solving it will give us the value of \( a \). After solving, we find:<br />\[ a = 4 \]<br /><br />Thus, the value of \( a \) is \( \boxed{4} \).