Questions: Evaluate the limit as x approaches 0 of (x^100 + 7x^2) / (x^50 - 4x^2)

Solution Steps

To evaluate the limit \(\lim _{x \rightarrow 0} \frac{x^{100}+7 x^{2}}{x^{50}-4 x^{2}}\), we need to analyze the behavior of the numerator and the denominator as \(x\) approaches 0. We can simplify the expression by factoring out the lowest power of \(x\) in both the numerator and the denominator.

We need to evaluate the limit: \[ \lim _{x \rightarrow 0} \frac{x^{100} + 7x^2}{x^{50} - 4x^2} \]

To simplify the expression, we factor out the lowest power of \(x\) in both the numerator and the denominator. The lowest power in the numerator is \(x^2\) and in the denominator is also \(x^2\): \[ \frac{x^{100} + 7x^2}{x^{50} - 4x^2} = \frac{x^2(x^{98} + 7)}{x^2(x^{48} - 4)} \]

We can cancel the common factor \(x^2\) from the numerator and the denominator: \[ \frac{x^{98} + 7}{x^{48} - 4} \]

Now, we evaluate the limit as \(x\) approaches 0: \[ \lim _{x \rightarrow 0} \frac{x^{98} + 7}{x^{48} - 4} \] As \(x \rightarrow 0\), \(x^{98} \rightarrow 0\) and \(x^{48} \rightarrow 0\). Therefore, the expression simplifies to: \[ \frac{0 + 7}{0 - 4} = \frac{7}{-4} = -\frac{7}{4} \]

Final Answer