∫ calculus

1. The problem asks to find the limit $$\lim_{x \to 3} \big(h(x) g(x)\big)$$. 2. To find this limit, we need the values of $$h(x)$$ and $$g(x)$$ near $$x=3$$ or their individual li

Problem: Identify a number $a$ for each description (a)–(d) and for the final shown graph, and explain why. 1. (a) Statement: $\lim_{x\to a} g(x)$ does not exist but $g(a)$ is defi

1. We are given a piecewise function $$g(x)$$ with points and limits described. 2. (a) Find $$a$$ where $$\lim_{x \to a} g(x)$$ does not exist but $$g(a)$$ is defined.

1. The problem states: For question 2, the limit $\lim_{x \to a} g(x)$ exists but $g(a)$ is not defined. 2. This means that as $x$ gets closer to $a$, the function values $g(x)$ ap

1. The problem asks to find values of $a$ where the function $g$ has different behaviors regarding the limit and value of $g(a)$. We analyze each part based on the graph descriptio

1. **State the problem:** We need to find the values of \( \lim_{x \to 5^+} h(x) \) and \( \lim_{x \to 5} h(x) \) from the given graph. 2. **Analyze the right-hand limit at x = 5:*

1. **State the problem:** We are given limits related to the function $h(x)$ and asked to determine specific limit values and function behavior based on a described graph. 2. **Giv

1. نبدأ بكتابة المعادلة المعطاة: $$\frac{d}{dx}(س) + س \frac{d}{dx}(-س) = س^3 + س^2 + س + 1$$ 2. نحسب المشتقات: المشتقة الأولى $$\frac{d}{dx}(س) = 1$$.

1. Discuss continuity of $f(x) = \begin{cases} \frac{x^2 - 9}{x - 3}, & x \neq 3 \\ 0, & x = 3 \end{cases}$ at $x=3$. Simplify the function for $x \neq 3$: $$\frac{x^2 - 9}{x - 3}

1. Problem: Find the following limits and function values for the piecewise graph of function $h$: (a) $\lim_{x \to -3^-} h(x)$

1. The problem asks us to find the partial derivative $\frac{\partial M}{\partial y}$ where $M=2\pi (3x + y - y e^{-x^2})$. 2. First, write $M$ explicitly and consider $x$ as a con

1. **State the problem:** We need to use linear approximation to estimate $\sin\left(\frac{9}{4}\right)$. 2. **Choose a point for approximation:** Since $\frac{9}{4} = 2.25$, which

1. **نبدأ بحل التكامل الأول: $$\int_0^{\frac{\pi}{2}} e^{\cos x} \sin(2x)\,dx.$$ 2. نستخدم تبديل الصيغ: \( \sin(2x) = 2\sin x \cos x \).

1. **Problem statement:** Use partial fraction decomposition to evaluate the integral $$\int \frac{8x^7 + 47x^6 + 98x^5 + 106x^3 + 100x^2 + 104x + 104}{(x - 1)(x + 2)^3 (x^2 + 2x +

1. The problem asks to solve two differential equations: a) $$\frac{dy}{dx} = \frac{1+x}{1+y}$$

1. Stating the problem: We need to find the limits of the function $f(x)$ as $x$ approaches infinity and negative infinity. 2. From the given information, as $x \to \infty$, the fu

1. Stating the problem: Evaluate the limit $$\lim_{x \to 0} \frac{\sqrt[3]{1 + cx - 1}}{x} = \lim_{x \to 0} \frac{\sqrt[3]{cx}}{x}.$$ 2. Simplify the expression inside the cube roo

1. The problem is to determine which of the functions \(f(x) = \frac{x^2 - 1}{x}\), \(f(x) = x \sqrt{x^2 + 1}\), or \(f(x) = x^2 - 1\) intersects the x-axis at \(x=1\) and is perpe

1. Differentiate the implicit equation $$7y^2 + \sin(3x) = 12 - y^4$$ with respect to $x$. Using implicit differentiation:

1. **State the problem parts:** (a) State the three conditions for continuity of function $f(x)$ at $x = x_0$.

1. Problem 31: Find relative asymptotes and study variation of $f(x) = x e^x$ 2. To find asymptotes, check behavior as $x \to \pm \infty$: