Value of $\lambda$ such that two limits are equal

Question

Referring to the figure below, find the value of $\lambda$ such that two limits are equal: $$2\left[{\lim_{x \to 0} f(x^3-x^2)}\right]=\lambda\left[{\lim_{x \to 0} f(2x^4-x^5)}\right]$$

Answer 1

When $x$ is close to $0$, the leading behavior is given by the lowest power. That is $x^2$ for the term on the left, and $x^4$ for the term on the right. In this case $-x^2+x^3$ is always negative in a small neighborhood of $0$, so the limit on the left is $3$. Similarly, $2x^4-x^5$ is greater than $0$, so the limit is $2$. Therefore $$2\cdot3=\lambda\cdot2$$