# Limit of parametric definite integral with discontinuity

How to find this limit?

$$I = \lim_{t \to 0+0} \int_1^3 \frac{\sin(tx)}{t} \sqrt{x^2 + t^2 + tx + 1} dx$$

The problem is that it is impossible to make the limit transition $$I = \int_1^3 \lim_{t \to 0+0} \frac{\sin(tx)}{t} \sqrt{x^2 + t^2 + tx + 1} dx,$$ because the integrand function has a discontinuity in $t = 0$, therefore, the previous statement is incorrect, generally speaking.

• what do you mean from $t \rightarrow 0+0$? – Khosrotash Feb 10 '15 at 17:56
• @darya-khosrotash, one-sided limit, $t \to 0$ and $t > 0$. – Nastya Koroleva Feb 10 '15 at 17:59

$$\frac{\sin(t\,x)}{t}=x\,\frac{\sin(t\,x)}{t\,x}=x\,\text{sinc}(t\,x),$$ and the function $$\text{sinc}(z)=\begin{cases}\dfrac{\sin z}{z} & z\ne0\\ 1 & z=0 \end{cases}$$ is continuous.