∫ calculus

1. We are asked to find the limit: $$\lim_{x \to 4} \frac{\sqrt{4+x} - \sqrt{2x}}{x-4}$$ 2. This is an indeterminate form of type $\frac{0}{0}$ because substituting $x=4$ gives $\f

1. **State the problem:** We want to find the derivative of the function $f(x) = \sqrt{4 - x}$ using the definition of the derivative. 2. **Recall the definition of the derivative:

1. The problem asks to identify all values of $x$ where the limit of the function $f(x)$ does not exist. 2. Limits do not exist at points where the function has vertical asymptotes

1. **State the problem:** Find the limit of the function $$\frac{x^2 - 1}{x - 1}$$ as $x$ approaches 1. 2. **Recall the formula and rules:** Direct substitution gives $$\frac{1^2 -

1. **Problem:** Test the limit, continuity, and differentiability of the function $$f(x) = \begin{cases} x^2, & x < 2 \\ 5, & x = 2 \\ x + 1, & x > 2 \end{cases}$$

1. **Problem statement:** Calculate the limit $$\lim_{n \to \infty} x_n$$ where $$x_n = \left(\frac{(n!)^3}{n^{3n} e^{-n}}\right)^{\frac{1}{n}}.$$ We are asked to solve part (e) us

1. **Problem statement:** Find the integral $$\int \ln(1+x) \, dx$$ using integration by parts. 2. **Formula and rule:** Integration by parts formula is $$\int u \, dv = uv - \int

1. The problem involves finding the integral using integration by parts where $u=\ln(x+2)$ and $dv=dx$. 2. Recall the integration by parts formula: $$\int u\,dv = uv - \int v\,du$$

1. **State the problem:** We need to find the integral $$\int \frac{3k^3 + 2k}{8 + k^2} \, dk.$$\n\n2. **Rewrite the integral:** Notice the numerator is a polynomial of degree 3 an

1. **Problem a:** Calculate the definite integral $$\int_0^1 \ln(x+2) \, dx$$ using integration by parts. 2. **Formula for integration by parts:** $$\int u \, dv = uv - \int v \, d

1. **Problem statement:** We want to evaluate the integral $$I = \int_0^{\infty} \frac{\ln(1 + x^2)}{1 + x^2} \, dx.$$\n\n2. **Key idea:** This integral involves the logarithm and

1. مسئله: انتگرال $$I_n = \int \frac{1}{x^n \sqrt{ax+b}} \, dx$$ را به صورت رابطه بازگشتی بیابید. 2. ابتدا قاعده انتگرال‌گیری جز به جز را یادآوری می‌کنیم: $$\int u \, dv = uv - \in

1. مسئله: انتگرال $$I_n = \int \frac{1}{x^n \sqrt{ax+b}} \, dx$$ را به صورت رابطه بازگشتی بیابیم. 2. ابتدا تعریف کنیم: $$I_n = \int \frac{1}{x^n \sqrt{ax+b}} \, dx$$ که در آن $$n \

1. Bài toán yêu cầu tính đạo hàm bậc 100 của hàm số $$f(x) = \frac{1}{x^2 + 4}$$. 2. Ta nhận thấy hàm số có dạng phân thức với mẫu là đa thức bậc 2.

1. **Problem:** Calculate the definite integral $$\int_0^3 x \, dx$$ 2. **Formula:** The integral of $$x$$ is $$\frac{x^2}{2}$$. For definite integrals, evaluate the antiderivative

1. Given the problem: Find the differential $dz$ for the function $$z = x^3 + x^2 y - x y^2 + 4 y^3.$$ 2. The formula for the total differential of a function $z = f(x,y)$ is:

1. **Problem:** Given the function $f(q) = hq^2 + mq + c$ with gradient function $4q + 8$ and a stationary value of $-3$, find $h$, $m$, and $c$. 2. **Step 1: Understand the gradie

1. **Problem:** Find $\frac{dy}{dx}$ using implicit differentiation for the equation $x^2 + y^2 = 25$. 2. **Formula:** Use the rule $\frac{d}{dx}[y^n] = n y^{n-1} \frac{dy}{dx}$ wh

1. **Problem Statement:** We need to sketch a function $f(x)$ defined on $\mathbb{R}$ that satisfies the following limit conditions:

1. **State the problem:** Evaluate the limit as $r$ approaches 0 of the expression $$\frac{5r^{-0.4} - 7r^{-\frac{3}{5}} + 3r^{-0.1} - 4r^{-\frac{3}{7}} - 8r^{-0.2}}{}$$

1. **State the problem:** Evaluate the limit as $r$ approaches 0 of the expression $$\frac{5r^{-0.4} - 7r^{-\frac{3}{5}} + 3r^{-0.1} - 4r^{-\frac{3}{7}} - 8r^{-0.2}}{}$$