Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE). $\newline$ $\lim _{x \rightarrow \infty} \frac{\sqrt{25 x^{4}-28 x^{2}}}{3 x+5 x^{2}}$ $\newline$ Answer:

View step-by-step help

Home

Math Problems

Precalculus

Find limits of polynomials and rational functions

Full solution

Q. Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE).

\newline

\lim _{x \rightarrow \infty} \frac{\sqrt{25 x^{4}-28 x^{2}}}{3 x+5 x^{2}}

\newline

Answer:

Simplify Expression: To find the limit of the given function as $x$ approaches infinity, we should first simplify the expression by dividing the numerator and the denominator by the highest power of $x$ in the denominator, which is $x^2$ .
Divide by $x^2$ : Divide both the numerator and the denominator by $x^2$ : $\lim_{x \to \infty}\left(\frac{\sqrt{25x^{4}-28x^{2}}}{3x+5x^{2}}\right) = \lim_{x \to \infty}\left(\frac{\sqrt{\frac{25x^{4}}{x^2} - \frac{28x^{2}}{x^2}}}{\frac{3x}{x^2} + \frac{5x^{2}}{x^2}}\right)$
Simplify Inside Limit: Simplify the expression inside the limit: \lim_{x \rightarrow \infty}\left(\frac{\sqrt{$25$x^{$4$}/x^$2$ - $28$x^{$2$}/x^$2$}}{$3$x/x^$2$ + $5$x^{$2$}/x^$2$}\right) = \lim_{x \rightarrow \infty}\left(\frac{\sqrt{$25$x^$2$ - $28$}}{$3$/x + $5$}\right)
Approaching Infinity: As \(x approaches infinity, the terms $\frac{3}{x}$ and $-\frac{28}{x^2}$ will approach $0$ . So we can simplify the expression further: $\newline$ $\lim_{x \to \infty}\left(\frac{\sqrt{25x^2 - 28}}{\frac{3}{x} + 5}\right) = \lim_{x \to \infty}\left(\frac{\sqrt{25x^2}}{5}\right)$
Square Root Simplification: The square root of $25x^2$ is $5x$ (since $x$ is approaching infinity, we consider only the positive square root): $\lim_{x \to \infty}\left(\frac{\sqrt{25x^2}}{5}\right) = \lim_{x \to \infty}\left(\frac{5x}{5}\right)$
Cancel Common Factor: Simplify the expression by canceling out the common factor of $5$ : $\lim_{x \rightarrow \infty}\left(\frac{5x}{5}\right) = \lim_{x \rightarrow \infty}x$
Limit as $x$ approaches Infinity: The limit of $x$ as $x$ approaches infinity is infinity: $\lim_{x \to \infty}x = \infty$