Animated Solution for Mathematics - Limits, Continuity and Differentiability: ${\lim }_{x\to 0}\csc x(\sqrt{2{\cos }^{2}x+3\cos x}-\sqrt{{\cos }^{2}x+\sin x+4})$ is:

We are tasked with evaluating the limit:

Substituting $x=0$ yields an indeterminate form of $\infty \cdot 0$. To resolve this, we rewrite the expression as a fraction:

This transformation results in the standard $\frac{0}{0}$ indeterminate form, signaling that we are ready to proceed with algebraic manipulation.

To eliminate the radicals, we multiply the numerator and the denominator by the conjugate expression:

Applying the identity $(a-b)(a+b)=a^2-b^2$, the numerator becomes:

Distributing the negative sign and simplifying, we obtain:

We recognize the quadratic component ${\cos }^{2}x+3\cos x-4$. By substituting $u=\cos x$, we factor the expression as $(u+4)(u-1)$, which gives:

The limit expression now takes the form:

For the first term, we utilize the identity $\frac{\cos x-1}{\sin x}=-\tan (\frac{x}{2})$. As $x\to 0$, this term approaches $0$:

For the second term, the $\sin x$ terms cancel, leaving:

Substituting $x=0$ into the denominator, we get $\sqrt{5}+\sqrt{5}=2\sqrt{5}$. Thus, the final result is:

$$L=-\frac{1}{2\sqrt{5}}$$