Math Problem Statement

Solve the limit problems: 1. \( \lim_{x \to +\infty} \frac{1 + 1/x}{x} \) 2. \( \lim_{x \to 0^-} \frac{\sin x + \tan x}{x} \) 3. \( \lim_{x \to +\infty} \frac{1}{\sqrt{x^2 + x} - x} \) 4. \( \lim_{x \to 0^+} \frac{1}{x^{1/2}} \) 5. \( \lim_{x \to 0} ((\sin x)^2 + (\cos x)^2) \) 6. \( \lim_{x \to +\infty} \left(\frac{1 + 1/x}{x} - 2x\right) \)

Solution

Thank you for providing the image. I'll go through each of the limit problems and provide a solution. Let's analyze them one by one.

Problem 3:

limx+(1+1/xx)\lim_{x \to +\infty} \left(\frac{1 + 1/x}{x}\right)

Solution for Problem 3:

As x+x \to +\infty, 1x0\frac{1}{x} \to 0. Therefore, 1+1x11 + \frac{1}{x} \to 1. 1+1xx1x0 as x+.\frac{1 + \frac{1}{x}}{x} \approx \frac{1}{x} \to 0 \text{ as } x \to +\infty. Answer: 0.


Problem 4:

limx0sinx+tanxx\lim_{x \to 0^-} \frac{\sin x + \tan x}{x}

Solution for Problem 4:

As x0x \to 0^-, both sinx\sin x and tanx\tan x can be approximated by xx, thus: sinx+tanxxx+xx=2.\frac{\sin x + \tan x}{x} \approx \frac{x + x}{x} = 2. Answer: 2.


Problem 5:

limx+1x2+xx\lim_{x \to +\infty} \frac{1}{\sqrt{x^2 + x} - x}

Solution for Problem 5:

Rationalize the denominator: 1x2+xxx2+x+xx2+x+x=x2+x+xx.\frac{1}{\sqrt{x^2 + x} - x} \cdot \frac{\sqrt{x^2 + x} + x}{\sqrt{x^2 + x} + x} = \frac{\sqrt{x^2 + x} + x}{x}. Then, as x+x \to +\infty, this simplifies to 1. Answer: 1.


Problem 6:

limx0+1x1/2\lim_{x \to 0^+} \frac{1}{x^{1/2}}

Solution for Problem 6:

As x0+x \to 0^+, x1/20x^{1/2} \to 0, making 1x1/2\frac{1}{x^{1/2}} diverge to ++\infty. Answer: ++\infty.


Problem 7:

limx0((sinx)2+(cosx)2)\lim_{x \to 0} ((\sin x)^2 + (\cos x)^2)

Solution for Problem 7:

Using the Pythagorean identity: (sinx)2+(cosx)2=1.(\sin x)^2 + (\cos x)^2 = 1. Answer: 1.


Problem 8:

limx+(1+1/xx2x)\lim_{x \to +\infty} \left(\frac{1 + 1/x}{x} - 2x\right)

Solution for Problem 8:

First term 0\to 0

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Math Problem Analysis

Mathematical Concepts

Calculus
Limits
Indeterminate Forms
L'Hôpital's Rule

Formulas

lim_{x \to \infty} (1/x) = 0
Pythagorean identity: (sin x)^2 + (cos x)^2 = 1
lim_{x \to 0} (1/x^{1/2}) = +∞

Theorems

L'Hôpital's Rule
Pythagorean Identity

Suitable Grade Level

College Level Calculus