Exercise Questions

All questions in this exercise are listed below. Click on a question to view its solution.

Key Concepts

This exercise focuses on the following concepts:

• Continuity at a point: ${\lim }_{x\to c}f(x)=f(c)$

• Types of discontinuities: removable, jump, infinite, and oscillating

• Continuity of polynomials and rational functions

• Domain restrictions and vertical asymptotes

• Continuity of trigonometric functions and special limits

• Piecewise functions and continuity at boundary points

• Intermediate Value Theorem (IVT) and existence of roots

• Solving for unknown parameters to ensure continuity

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Important Formulas

Below are the key formulas used in this exercise:

Continuity Condition: ${\lim }_{x\to c}f(x)=f(c)$

Piecewise Continuity at Boundary $c$: ${\lim }_{x\to c^{-}}f(x)={\lim }_{x\to c^{+}}f(x)=f(c)$

Special Trigonometric Limit: ${\lim }_{x\to 0}\frac{\sin x}{x}=1$

Intermediate Value Theorem: If $f$ is continuous on $[a,b]$ and $k$ is between $f(a)$ and $f(b)$, then $\exists c\in (a,b)$ such that $f(c)=k$.

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Summary

This exercise develops proficiency in analyzing function continuity across algebraic, transcendental, and piecewise functions. Problems 1-14 emphasize identifying discontinuities by examining domain restrictions, singularities (where denominators vanish), and behavior at critical points—particularly distinguishing between removable discontinuities (holes) and infinite discontinuities (asymptotes).

Problems 15-18 require solving for unknown parameters ($m$, $n$) to enforce continuity, demanding careful calculation of one-sided limits and function values at boundary points. Problem 19 applies the Intermediate Value Theorem to prove existence of solutions within intervals, while Problem 20 examines the theoretical construct of a function discontinuous everywhere. Key strategies include algebraic factorization to simplify rational expressions, evaluating limits of indeterminate forms, and systematic verification of the three continuity conditions.