Math Problem Statement

Find the limits for the following expressions: 1. \( \lim_{x \to \infty} \frac{2 + 3x}{x^2 + 1} \) 2. \( \lim_{x \to 2^-} \frac{x^3 - |x|}{x^3 - 2x^2 - x + 2} \) 3. \( \lim_{x \to \infty} \frac{3x^3 + 2x + 1}{1 - \sqrt{x^2}} \) 4. \( \lim_{x \to 0} \left( \sqrt{x^2 + x} - \sqrt{x^2 - x} \right) \)

Solution

It looks like you've uploaded an image containing limit problems. I will transcribe and solve each of the limits step by step.

Problem 5.1

limx2+3x(x2+1)\lim_{x \to \infty} \frac{2 + 3x}{(x^2 + 1)}

Solution:

For large values of xx, the higher degree terms dominate. So, we approximate the expression by considering the dominant terms:

limx2+3xx2+1limx3xx2\lim_{x \to \infty} \frac{2 + 3x}{x^2 + 1} \approx \lim_{x \to \infty} \frac{3x}{x^2}

This simplifies to:

limx3x=0\lim_{x \to \infty} \frac{3}{x} = 0

Thus, the limit is: 0\boxed{0}


Problem 5.2

limx2x3xx32x2x+2\lim_{x \to 2^-} \frac{x^3 - |x|}{x^3 - 2x^2 - x + 2}

Solution:

First, observe that as x2x \to 2^-, we approach from the left, so for values of xx near 2, we simplify the expression.

Let's directly substitute x=2x = 2:

For x|x| (since xx is approaching from the left, x<0x < 0):

For now, substituting x=2x = 2 in, and checking for undefined behavior.

Let me know

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Limits
Calculus
Infinite Limits
Piecewise Functions

Formulas

Limit of a rational function as x approaches infinity
Absolute value function in limits
Radical expressions in limits

Theorems

Limits involving infinity
Piecewise limits
Squeeze theorem (for limit problems involving square roots)

Suitable Grade Level

University level Calculus or Advanced High School (AP Calculus)