∫ calculus

1. **State the problem:** We need to find the limit $$\lim_{x \to 3} \frac{g(x)}{f(x)}$$ where functions $g$ and $f$ are given graphically. 2. **Analyze the behavior of $g(x)$ near

1. **State the problem:** Find the limit $$\lim_{x\to -1} \frac{12x^3 + 12x^2}{x^4 - x^2}$$. 2. **Analyze the expression:** The limit is a rational function. We first check if dire

1. **State the problem:** We need to find the limit $$\lim_{x \to \frac{\pi}{4}} \frac{\cos(2x)}{\cos(x) - \sin(x)}$$. 2. **Recall formulas and rules:**

1. **State the problem:** We need to find the limit $$\lim_{x\to 3} \frac{x-3}{2-\sqrt{x+1}}.$$\n\n2. **Identify the issue:** Direct substitution gives $$\frac{3-3}{2-\sqrt{3+1}}=\

1. **Problem Statement:** We are given a piecewise function: $$f(x)=\begin{cases} \frac{2}{x^2} & \text{for } x \leq -1 \\\ \frac{x+3}{\cos(x+1)} & \text{for } -1 < x < \frac{\pi -

1. **State the problem:** Evaluate the limit $$\lim_{x \to 1} \frac{|x-1|}{1-x}$$. 2. **Recall the definition of absolute value:**

1. **State the problem:** We need to find the limit $$\lim_{\theta \to 0} \frac{1 - \cos(\theta)}{2 \sin^2(\theta)}$$. 2. **Recall important formulas and rules:**

1. **State the problem:** Find the limit $$\lim_{x\to 1} \frac{2x}{x^2 - 7x + 6}$$. 2. **Recall the formula and approach:** To find the limit of a rational function as $x$ approach

1. The problem asks if Hayley's suggestion to use the functions \(g(x)=e^x\) and \(h(x)=e^{-x}\) to apply the squeeze theorem for function \(f(x)\) near \(x=0\) is correct. 2. The

1. **State the problem:** We need to find the limit $$\lim_{x\to 3} \frac{\sqrt{2x-5}-1}{x-3}.$$\n\n2. **Identify the indeterminate form:** Substituting $x=3$ directly gives $$\fra

1. **State the problem:** We are given a piecewise function $$f(x)=\begin{cases}\ln(-x)+3 & \text{for } x < -3 \\\\ \ln(-x+3) & \text{for } -3 \leq x < 3 \end{cases}$$

1. **State the problem:** We need to find the limit $$\lim_{x\to 2} \frac{x^4 + 3x^3 - 10x^2}{x^2 - 2x}.$$\n\n2. **Check direct substitution:** Substitute $x=2$ into numerator and

1. The problem asks which of the functions \(g(x) = \sqrt[5]{x}\) and \(h(x) = \sqrt[3]{x}\) are continuous for all real numbers. 2. Recall that the \(n\)-th root function \(f(x) =

1. **State the problem:** Find the limit as $n$ approaches $+\infty$ of the expression $$\frac{3n^2 - n - 1}{(2n - 1)^2}.$$\n\n2. **Recall the formula and rules:** When evaluating

1. **Problem:** Determine if the series $$\sum_{n=1}^\infty \frac{1}{n^2}$$ converges or diverges using the Integral Test. 2. **Integral Test conditions:** The function $$f(x) = \f

1. **State the problem:** Find the absolute minimum value of the function $f(x) = \cos(x)$ on the interval $[0, \pi]$. 2. **Recall the function and interval:** The cosine function

1. **Stating the problem:** We have a function $f : \mathbb{R} \to \mathbb{R}$ defined by $$f(x) = \sqrt[3]{x^2}(3x - 7) = x^{\frac{2}{3}}(3x - 7).$$ We want to find values $a$ and

1. **State the problem:** We need to find the intervals where the function $f(x) = |9 - x^2|$ is decreasing. 2. **Understand the function:** The function is the absolute value of $

1. **State the problem:** The radius $r$ of a sphere is increasing at a rate of $\frac{dr}{dt} = 5$ cm/s. We need to find the rate at which the surface area $S$ is increasing when

1. **State the problem:** We need to find the first derivative $f'(x)$, the second derivative $f''(x)$, and then evaluate both at $x=1$ for the function $$f(x) = \frac{2x}{x^2 + 1}

1. **State the problem:** We are given the function $$f(x) = \frac{2x}{x^2 + 1}$$ and need to find its first derivative $$f'(x)$$, second derivative $$f''(x)$$, and then evaluate t