Math Problem Statement

Solve with complete and proper solutions.

Solve with complete and proper solutions.

Solution

Let's work through each of the problems in the image step by step. I'll provide detailed solutions for all of them.

Problem 1:

lim⁡x→1x2+4x−5\lim_{x \to 1} \sqrt{x^2 + 4x - 5}

Solution:

  1. Substitute x=1x = 1 into the expression: [ x^2 + 4x - 5 = 1^2 +

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Math Problem Analysis

Mathematical Concepts

Limits
Trigonometric Limits
Algebraic Limits
L'Hôpital's Rule
Factoring

Formulas

lim (x → a) f(x) = L
L'Hôpital's Rule: lim (x → a) f(x)/g(x) = lim (x → a) f'(x)/g'(x) if the form is 0/0 or ∞/∞
Trigonometric Identity: 1 - cos(x) ≈ x^2/2 as x → 0
Factorization Techniques
Rationalization

Theorems

L'Hôpital’s Rule
Squeeze Theorem
Fundamental Limit Theorems

Suitable Grade Level

Undergraduate Calculus / Advanced High School (Grades 11-12)

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