∫ calculus

1. Soal 1: Diketahui fungsi $$f(x) = \begin{cases} 3 - (x - 2), & x < 3 \\ 2, & x > 3 \end{cases}$$

1. Let's start by stating what partial derivatives are. 2. Partial derivatives measure how a multivariable function changes as we vary one variable while keeping the others constan

1. **Problem statement:** Determine the following limits based on the described graph of the function $f$. 2. **(a) lim $x\to1^-$ $f(x)$:** The graph shows a solid dot at $(1,2)$ f

1. The problem is to find the derivative of a quadratic equation and show the steps for its derivation. 2. Consider the quadratic function $f(x) = ax^2 + bx + c$ where $a$, $b$, an

1. The problem is to find the derivative of the function $$f(x) = e^{\frac{4x - x^5}{7x^2}}$$

1. The problem is to find the limit \( \lim_{x \to 2} \frac{x^2 - 8}{x - 2} \). 2. First, substitute \( x = 2 \) directly to check for an indeterminate form:

1. **Stating the problem:** We want to find the limit

1. **State the problem:** Find the derivative of the function $$ f(x) = e^{\frac{3x - x^5}{6x^2}} $$

1. **Problem statement:** (a) Find the domain of the function $$g(x) = \sqrt{1 - \sin(x)} \tan \left|\frac{3x}{x^2 + 1}\right|.$$

1. **Stating the problem:** We want to find the limit of the sequence $$a_n=\frac{(n!)^{\frac{1}{n}}}{n}$$ as $$n$$ tends to infinity. 2. **Rewrite the expression:**

1. **State the problem:** We want to evaluate the limit $$\lim_{x \to 25} \frac{\sqrt{x} - 5}{x - 25}$$. 2. **Recognize the indeterminate form:** Direct substitution gives $$\frac{

1. **State the problem:** We need to evaluate the limit $$\lim_{x\to 4} \frac{\sqrt{x+5} - 3}{x - 4}$$ and round the result to the nearest thousandth. 2. **Direct substitution:** S

1. We are asked to find the limit: $$\lim_{x\to 0} \frac{\sin^3(4x)}{9x^2}$$. 2. Recall the standard limit: $$\lim_{t\to 0} \frac{\sin t}{t} = 1$$.

1. **State the problem:** Evaluate the limit $$\lim_{x \to 0} \frac{\sin x}{5x}$$. 2. **Recall the standard limit:** We know that $$\lim_{x \to 0} \frac{\sin x}{x} = 1$$.

1. The problem is to evaluate the limit $$\lim_{x \to 0}$$. 2. However, the expression or function for which the limit is to be found is missing.

1. The problem asks to find the limit $$\lim_{x \to \pi} \frac{2 \sin x}{x - \pi}.$$\n\n2. Observe that this is a limit which resembles the form $$\lim_{x \to a} \frac{f(x)}{x - a}

1. We are asked to evaluate the limit $$\lim_{n \to \infty} \left(1 - \arctan\left(\frac{1}{2}\right)\right)^n$$. 2. First, recall that $$\arctan\left(\frac{1}{2}\right)$$ is the a

1. Problem: Evaluate the limit $$\lim_{x \to -4} \frac{2x^2 - 4x - 48}{x^2 + 6x + 8}$$. 2. First, plug in $x = -4$ to check if the limit is directly evaluable:

1. **State the problem:** Evaluate the limit $$\lim_{x \to 6} \frac{x - 6}{x^2 - 4x - 12}$$ and simplify the answer. 2. **Factor the denominator:**

1. **State the problem:** Evaluate the limit $$\lim_{x \to 3} \frac{-3x^2 + 6x + 9}{x^2 - 9}$$ and simplify your answer. 2. **Substitute $x=3$ directly:** Calculate numerator and d

1. **State the problem:** Evaluate the limit $$\lim_{x \to 2} \frac{3x^3 - 6x^2}{x^2 - 2x}.$$\n\n2. **Substitute $x=2$ to check if direct evaluation works:**\nCalculate numerator: