Step-by-Step Solutions to Limit Problems Including Trigonometric and Algebraic Limits

Math Problem Statement

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:

$\lim_{x \to 1} \sqrt{x^2 + 4x - 5}$

Solution:

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

Ask a new question for Free

By Image

Drop file here or Click Here to upload

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)

Related Recommendation

Solving Limit Problems with Indeterminate Forms and Trigonometric Functions

Evaluation of Limits Involving Trigonometric and Exponential Functions

Solving Limits Using L'Hôpital's Rule: π/2 and π/4 Examples

Solving Limits: Examples and Solutions

Step-by-Step Solutions to Limit Problems in Calculus