$\lim_{x \rightarrow 0} \csc x \left(\sqrt{2 \cos^2 x + 3 \cos x} - \sqrt{\cos^2 x + \sin x + 4}\right)$ is:

Step 1: Rewrite the expression

First, we rewrite the given expression by replacing $\csc x$ with $\frac{1}{\sin x}$. This makes the expression a fraction, which is often easier to work with when evaluating limits, especially when it involves indeterminate forms.

Step 2: Check for indeterminate form

Next, we substitute $x=0$ into the expression to check for an indeterminate form. Since $\cos 0 = 1$ and $\sin 0 = 0$, we find that the expression evaluates to $\frac{0}{0}$, which is an indeterminate form. This indicates that we can use L'Hôpital's Rule or algebraic manipulation.

Step 3: Multiply by the conjugate

To resolve the indeterminate form, we multiply the numerator and the denominator by the conjugate of the numerator. The conjugate of $(\sqrt{A} - \sqrt{B})$ is $(\sqrt{A} + \sqrt{B})$. This eliminates the square roots in the numerator, simplifying the expression.

Step 4: Simplify the numerator

We expand and simplify the numerator. Combining like terms, we get $\cos^2 x + 3 \cos x - \sin x - 4$. The denominator remains the product of $\sin x$ and the conjugate term.

Step 5: Apply L'Hôpital's Rule

Since substituting $x=0$ still yields $\frac{0}{0}$ for the simplified expression, we apply L'Hôpital's Rule. We differentiate the numerator and the denominator with respect to $x$. The derivative of the numerator is $-2 \cos x \sin x - 3 \sin x - \cos x$. The derivative of the denominator is more complex, involving the product rule and chain rule.

Step 6: Evaluate the limit after L'Hôpital's Rule

Now, we substitute $x=0$ into the differentiated expression. Many terms involving $\sin x$ become zero. The numerator simplifies to $-1$. The denominator simplifies to $1 \cdot (\sqrt{5} + \sqrt{5}) = 2\sqrt{5}$.

More from this chapter

• Evaluate the limit as x approaches 0 of (e^(x²) - cos(x)) / (x²).
• Evaluate the limit as n approaches infinity of sum from k=1 to n of n / (n² + k²).
• Evaluate: lim(x->0) (sin(5x) - sin(3x)) / (sin(x)).
• Find the derivative of the function given by f(x)=sin (x²).
• Compute the derivative of 6 x^(100)-x^(55)+x.
• Find the derivative of f(x) = x³ - 3x² + 2x - 5 with respect to x.
• Evaluate: (i) lim _x → 1 (x^(15)-1)/(x^(10)-1) (ii) lim _x → 0 (√(1+x)-1)/(x)
• The value of lim_n → ∞ ( Σ_k=1^n (k³ + 6k² + 11k + 6)/((k+3)!)) is:

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