Evaluate $\displaystyle\lim_{x \to 0^{-}} \frac{\sqrt{x^2 + x + 4} - 2}{\ln(1 + x + x^2) - x}$

Question

I have been trying to evaluate

\begin{equation*} \lim_{x \to 0^{-}} \frac{\sqrt{x^2 + x + 4} - 2}{\ln(1 + x + x^2) - x}. \end{equation*}

We have \begin{equation*} \frac{\sqrt{x^2 + x + 4} - 2}{\ln(1 + x + x^2) - x} = \frac{x^2 + x}{\ln(1 + x + x^2) - x} \cdot \frac{1}{\sqrt{x^2 + x + 4} + 2} \quad \text{for all}\ x \in \mathbb{R}, \end{equation*} so I think my exercise boils down to \begin{equation*} \lim_{x \to 0^{-}} \frac{x^2 + x}{\ln(1 + x + x^2) - x}, \end{equation*} and this limit equals $-\infty$ by De L'Hôpital's Theorem.

Can I evaluate this limit without De L'Hôpital's Theorem?

Answer 1

Define $f(y) = \ln(1+y)$. We know that, $f'(0) = 1$. Thus $$\lim_{y\to 0} \frac{\ln(1+y)}{y} = \lim_{y\to 0} \frac{f(y)-f(0)}{y-0} = f'(0) = 1 \hspace{1cm}(1)$$ Now, consider the limit $$\lim_{x\to 0^{-}}\frac{\ln(1+x+x^{2})-x}{x^{2}+x} \hspace{1cm} (2)$$ Note that $x^{2}+x = (x+\frac{1}{2})^{2}-\frac{1}{4}$. Take $u = x+\frac{1}{2}$. If $x \to 0^{-}$ than $u \to \frac{1}{2}^{-}$. Now, (2) becomes $$ \lim_{u\to \frac{1}{2}^{-}}\frac{\ln(1+u^{2}-\frac{1}{4})-\bigg{(}u-\frac{1}{2}\bigg{)}}{u^{2}-\frac{1}{4}} = \lim_{u\to \frac{1}{2}^{-}}\frac{\ln(1+(u+1/2)(u-1/2))-\bigg{(}u-\frac{1}{2}\bigg{)}}{(u+1/2)(u-1/2)}$$ We break into two limits. The second one is $$ \lim_{u\to \frac{1}{2}^{-}}\frac{-\bigg{(}u-\frac{1}{2}\bigg{)}}{\bigg{(}u+\frac{1}{2}\bigg{)}\bigg{(}u-\frac{1}{2}\bigg{)}} = -1$$ The first one is $$\lim_{u\to \frac{1}{2}^{-}} \frac{\ln(1+(u+1/2)(1-1/2))}{(u+1/2)(u-1/2)}$$ is equivalent to the limit given by (1). Thus, the limit (2) goes to zero from the left.

Answer 2

Hint: $$ (1\pm f(x))^n \approx 1\pm nf(x) $$ $$ \ln(1\pm f(x)) \approx \pm f(x)$$ for $ f(x) \approx 0$ as $x\to 0$.

Answer 3

For small $x$ so $x^2+x$ is also small,$$\sqrt{x^2+x+4}-2=2\left(\sqrt{1+\frac14(x^2+x)}-1\right)\approx\frac14(x^2+x)\approx\frac14 x$$while$$\ln(1+x+x^2)-x\approx x(1+x)-\frac12 x^2(1+x)^2-x\approx\frac12 x^2,$$so the ratio $\approx\frac{1}{2x}\to-\infty$ as $x\to0^-$.