How can I calculate this limit without using exponential or logs as I found on internet?
Solution 1:
$$\lim_{x \to \frac{1}{4}} \frac{x^2}{\left(2\sqrt{x} - 1\right)^2}$$
$$\implies \frac{(\frac{1}{4})^2}{\left(2\sqrt{\frac{1}{4}} - 1\right)^2}$$
$$\implies \frac{\frac{1}{16}}{\left(2\cdot\frac{1}{2} - 1\right)^2}$$
$$\implies \frac{\frac{1}{16}}{\left(1 - 1\right)^2}$$
$$\implies \frac{\frac{1}{16}}{0^+}$$
$$\implies \frac{1}{0^+}$$
This quantity will appoach $+\infty$.
See the graph
