Question

Compute the following limits and find the integrals 1) $\lim_{x \to \infty} \frac{x^2(1+sin^2(x))}{(x+sin(x))^2}$ 2) $\lim_{x \to y} \frac{x^{2024} - y^{2024}}{x - y}$ 3) $\int \frac{dx}{\sqrt{x} + \sqrt[3]{x}}$ 4) $\int Log(cos(x)) tan(x) dx

          Compute the following limits and find the integrals
1) $\lim_{x \to \infty} \frac{x^2(1+sin^2(x))}{(x+sin(x))^2}$
2) $\lim_{x \to y} \frac{x^{2024} - y^{2024}}{x - y}$
3) $\int \frac{dx}{\sqrt{x} + \sqrt[3]{x}}$
4) $\int Log(cos(x)) tan(x) dx

Compute the following limits and find the integrals
1) limx →∞(x^2(1+sin^2(x)))/((x+sin(x))^2)
2) limx → y(x^2024 - y^2024)/(x - y)
3) ∫(dx)/(√(x) + √(x))
4) ∫ Log(cos(x)) tan(x) dx

Added by Gregorio L.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sahil Kumar

02/20/2022

Step 1

Step 1: For the first limit, we can divide both numerator and denominator by $x^2$ to get: $$\lim_{x \to \infty} \frac{1+sin^2(x)}{(1+\frac{sin(x)}{x})^2}$$ Since $-1 \le sin(x) \le 1$, we have: $$\lim_{x \to \infty} \frac{sin(x)}{x} = 0$$ Therefore, the limit Show more…

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Compute the following limits and find the integrals 1) $$\lim_{x \to \infty} \frac{x^2(1+sin^2(x))}{(x+sin(x))^2}$$ 2) $$\lim_{x \to y} \frac{x^{2024}-y^{2024}}{x-y}$$ 3) $$\int \frac{dx}{\sqrt{x}+\sqrt[3]{x}}$$ 4) $$\int Log(cos(x))tan(x)dx$$