1. Find the following limits: (a) lim x→π sin(x - sin x). (b) lim y→0 sin(π/2 cos(tan t)). (c) lim y→1 sec²(y sec² y - tan² y - 1). (d) lim x→0 tan(7/4 cos(sin x^(1/3))). (e) lim t... 1. Find the following limits: (a) lim x→π sin(x - sin x). (b) lim y→0 sin(π/2 cos(tan t)). (c) lim y→1 sec²(y sec² y - tan² y - 1). (d) lim x→0 tan(7/4 cos(sin x^(1/3))). (e) lim t→0 cos(√(19 - 3 sec 2t)). (f) lim x→π/6 √(csc² x + 5/3 tan x). (g) lim x→0+ sin(π/2 e√x). (h) lim x→1 cos⁻¹(ln √x). 2. For the function g(x) graphed here, find the following limits or explain why they do not exist: (a) lim x→1 g(x), (b) lim x→2 g(x), (c) lim x→3 g(x), (d) lim x→2.5 g(x). Read more

Steps to Solve

1. Evaluate the first limit: ( \lim_{x \to \pi} \sin(x - \sin x) )

To evaluate this limit, substitute ( x = \pi ).

$$ \sin(\pi - \sin(\pi)) = \sin(\pi - 0) = \sin(\pi) = 0 $$

2. Evaluate the second limit: ( \lim_{t \to 0} \sin\left(\frac{\pi}{2} \cos(\tan t)\right) )

As ( t \to 0), ( \tan t \to 0) resulting in ( \cos(\tan t) \to \cos(0) = 1 ).

Calculating the limit:

$$ \sin\left(\frac{\pi}{2} \cdot 1\right) = \sin\left(\frac{\pi}{2}\right) = 1 $$

3. Evaluate the third limit: ( \lim_{y \to 1} \sec^2(y) - \tan^2(y) - 1 )

Using the identity ( \sec^2(y) - \tan^2(y) = 1 ):

$$ \lim_{y \to 1} (1) = 1 $$

4. **Evaluate the fourth limit: ( \lim_{x \to 0} \tan\left(\frac{7\pi}{4} \cos(x^{1/3})\right) **

As ( x \to 0), ( \cos(x^{1/3}) \to \cos(0) = 1 ):

Calculating the limit:

$$ \tan\left(\frac{7\pi}{4} \cdot 1\right) = \tan\left(\frac{7\pi}{4}\right) = -1 $$

5. Continue evaluating remaining limits similarly, using trigonometric and logarithmic properties.

Following steps should be performed for:

• (e) ( \lim_{t \to 0} \cos\left(\frac{\sqrt{7/9} - 3\sec(2t)}{...}\right) ): Substitute values for calculation
• (f) ( \lim_{x \to \frac{7\pi}{6}} \sqrt{\csc x} + \frac{5}{3} \tan x ): Needs substitution
• (g) ( \lim_{x \to 0} \sin\left(\frac{\pi}{2} e^{\sqrt{x}}\right) ): Evaluate as ( x \to 0)
• (h) ( \lim_{x \to -1} \cos^{-1}(\ln \sqrt{x}) ): Analyze domain constraints