To analyze the equations in the provided image:

1. First, we will prove the first equation step-by-step.
2. Then, using the result of the first equation, we will solve the second equation.

Let’s rewrite the two equations seen in the image clearly:

1st Equation (Prove):

$\lim_{x \to \frac{\pi}{4}} \frac{\frac{4}{3} - \frac{1}{3} \tan(x)}{\frac{1}{x - \frac{\pi}{4}}}$

2nd Equation (Solve):

$\lim_{x \to 2} \frac{\frac{1}{x} + 0.3 \tan(x)}{\frac{2}{x}} = e$

Let’s first prove the first equation step by step. Then, we will use the result for the second part. sympy import symbols, tan, limit, pi

Defining symbols

x = symbols('x') expr1 = (4/3 - (1/3)*tan(x)) / (1/(x - pi/4))

Computing the limit for the first equation

limit_expr1 = limit(expr1, x, pi/4) limit_expr1The result of the first limit is:

$\lim_{x \to \frac{\pi}{4}} \frac{\frac{4}{3} - \frac{1}{3} \tan(x)}{\frac{1}{x - \frac{\pi}{4}}} = 0$

Proof of the First Equation:

1. Observe that as $x \to \frac{\pi}{4}$, the numerator $\frac{4}{3} - \frac{1}{3} \tan(x)$ approaches a constant value.
2. The denominator $\frac{1}{x - \frac{\pi}{4}}$ grows infinitely large as $x$ approaches $\frac{\pi}{4}$.
3. This results in the limit evaluating to $0$, as verified above.

Now, let’s use this result to solve the second equation. Defining the second equation expr2 = ((1/x) + 0.3 * tan(x)) / (2/x)

Computing the limit for the second equation

limit_expr2 = limit(expr2, x, 2) limit_expr2The result of the second limit is:

$\lim_{x \to 2} \frac{\frac{1}{x} + 0.3 \tan(x)}{\frac{2}{x}} = 0.3 \tan(2) + \frac{1}{2}$

Steps to Solve the Second Equation:

1. Numerator: $\frac{1}{x} + 0.3 \tan(x)$ As $x \to 2$, substitute values into each term.

2. Denominator: $\frac{2}{x}$ As $x \to 2$, this simplifies to $\frac{2}{2} = 1$.

3. The final result combines both parts and simplifies to: $0.3 \tan(2) + \frac{1}{2}$

Let me know if you’d like a detailed explanation of any part!

Here are 5 follow-up questions for practice:

1. How would the result change if the numerator in the second equation were squared?
2. What is the derivative of the numerator in the second equation?
3. Can you generalize the second equation to other values of $x$?
4. How does the value of $\tan(x)$ affect the limit as $x \to 2$?
5. What happens if the denominator of the second equation approaches zero?

Tip: Always verify the domain of functions like $\tan(x)$ to avoid undefined regions!