∫ calculus

1. The problem is to find the limit of a function or expression as the variable approaches a certain value. 2. Since the user only wrote "Lim" without specifying the function or th

1. The problem is to find the limit $$\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$$. 2. Notice that directly substituting $x = 3$ gives $$\frac{3^2 - 9}{3 - 3} = \frac{9 - 9}{0} = \frac{0

1. The problem is to find the limit $$\lim_{x \to 3} \frac{x^2 - 9x}{3}$$. 2. First, substitute $x = 3$ directly into the expression:

1. The problem is to find the limit $$\lim_{x \to 3} x^2 - 9x - 3$$. 2. First, substitute $x = 3$ directly into the expression:

1. **State the problem:** We have a storage tank with 10,000 litres of chemical leaking at a rate given by the derivative of the amount spilled, $$f'(t) = 400e^{-0.01t}$$ where $$t

1. **Problem statement:** Consider the function $f$ defined on $]0; +\infty[$ by $$f(x) = x - \frac{\ln x}{x}.$$

1. **State the problem:** We need to evaluate the definite integral $$\int_{-\pi}^{\frac{\pi}{2}} 3 \sin(x) + \cos(3x) \, dx.$$\n\n2. **Split the integral:** Use linearity of integ

1. **State the problem:** We are given the function $f(x) = \frac{e^x}{x^2}$ and need to find its derivative $f'(x)$. 2. **Identify the rule to use:** Since $f(x)$ is a quotient of

1. The problem is to find the derivative of the function $$f(x) = e^{x} x^{2}$$. 2. We recognize that $$f(x)$$ is a product of two functions: $$u(x) = e^{x}$$ and $$v(x) = x^{2}$$.

1. **State the problem:** We are given the equation $$y\sqrt{x+1} = 4$$ and asked to find $$\frac{dy}{dx}$$ using implicit differentiation. 2. **Rewrite the equation:** The equatio

1. **State the problem:** We have a circular wetted area with area $A$ expanding at a rate of $\frac{dA}{dt} = 4$ mm$^2$/s. We want to find how fast the radius $r$ is expanding, i.

1. **State the problem:** We are given that the volume $V$ of a sphere is increasing at a rate of $\frac{dV}{dt} = 7$ cm³/s. We need to find the rate of change of its surface area

1. The problem is to find the derivative of $y$ with respect to the independent variable $t$. 2. To proceed, we need the explicit function $y(t)$ or the relationship between $y$ an

1. **State the problem:** Find the absolute extreme values of the function $$f(x) = \ln(x + 2) + \frac{1}{x}$$ on the interval $$[1, 10]$$. 2. **Find the derivative:** To locate cr

1. **State the problem:** We need to find the derivative $\frac{dy}{dt}$ where $y = \left(e^{\cos(t+9)}\right)^4$. 2. **Rewrite the function:** We can write $y$ as $y = e^{4\cos(t+

1. **Evaluate the limits involving absolute value:** Given the function $$f(x) = \frac{1}{x} - \frac{1}{|x|}$$

1. The problem asks us to evaluate the infinite series \(\sum_{k=1}^{\infty} \left(\frac{1}{\sqrt{k+1}} - \frac{1}{\sqrt{k+2}}\right)\) or determine if it diverges. 2. Notice that

1. **State the problem:** We want to find a formula for the partial sum $$S_n = \sum_{k=1}^n \left( \frac{1}{\sqrt{k+1}} - \frac{1}{\sqrt{k+2}} \right)$$ and then calculate the lim

1. **State the problem:** We are given the sequence $$S_n = \frac{\sin((n+1)\pi)}{12n+11}$$ and asked to find the limit as $$n \to \infty$$. 2. **Analyze the numerator:** Note that

1. **State the problem:** We want to analyze the infinite series $$\sum_{k=0}^\infty \left( \sin\left( \frac{(k+1)\pi}{12k+11} \right) - \sin\left( \frac{k\pi}{12k-1} \right) \righ

1. **State the problem:** We want to find the value of the infinite sum $$\sum_{k=0}^\infty \left( \sin \left( \frac{(k+1)\pi}{12k+11} \right) - \sin \left( \frac{k\pi}{12k-1} \rig