Find the value of the constant $C$ for which the following integral converges and evaluate the integral for this value of $C$
$$\int_{1}^{\infty}\left[\frac{C}{x+2} - \frac{1}{\sqrt{x^2 +4}} \right]dx = \lim_{t\rightarrow\infty}\int_{1}^{t}\left[\frac{C}{x+2} - \frac{1}{\sqrt{x^2 +4}} \right]dx = \lim_{t\rightarrow\infty} \left(\sinh^{-1}(\frac{t}{2}) - C\ln(t+2)- \text{constant}\right)$$
By taking $\ln(t+2)$ common factor $$ \lim_{t\rightarrow\infty} \ln(t+2)\left(\frac{\sinh^{-1}(\frac{t}{2})}{\ln(t+2)} - C\right)- \text{constant}$$
$$ \text{Since} \lim_{t\rightarrow\infty} \frac{\sinh^{-1}(\frac{t}{2})}{\ln(t+2)} = 1 \text{ then C has to be equal 1} $$
and this is where I am confused $$ \lim_{t\rightarrow\infty} \ln(t+2) = \infty $$
and when we multiply this by $0$, we get undefined, so why should the $C$ be $1$. It is stated that if $C$ was anything but $1$ the answer would be $\infty$