# **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: - **Graphical Interpretation of Limits:** - Observing the behavior of $f(x)$ as $x$ approaches a value $c$ from both the left ($c^-$) and the right ($c^+$). - **One-Sided Limits and Existence:** - A limit $\lim_{x \rightarrow c} f(x)$ exists if and only if $\lim_{x \rightarrow c^-} f(x) = \lim_{x \rightarrow c^+} f(x)$. - **Algebraic Limit Techniques:** - **Direct Substitution:** The first step for continuous functions. - **Factoring:** Used to eliminate indeterminate forms like $0/0$ (e.g., $x^2-a^2$ or $x^3 \pm a^3$). - **Rationalization:** Multiplying by the conjugate to simplify expressions with radicals. - **Special Trigonometric Limits:** - Understanding the fundamental theorem $\lim_{x \rightarrow 0} \frac{\sin x}{x} = 1$. --- # **Important Formulas** Below are the key formulas used in this exercise: | Description | Formula | | :--- | :--- | | **Sum of Cubes** | $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ | | **Difference of Cubes** | $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ | | **Difference of Squares** | $a^2 - b^2 = (a - b)(a + b)$ | | **Trigonometric Identity** | $\lim_{x \rightarrow 0} \frac{\sin x}{x} = 1$ | | **Tangent Identity** | $\lim_{x \rightarrow 0} \frac{\tan x}{x} = 1$ | | **Limit of Sine (General)** | $\lim_{x \rightarrow 0} \frac{\sin ax}{ax} = 1$ | --- # **Summary** This exercise covers the foundational methods for evaluating limits. It begins with **graphical analysis**, highlighting that for a limit to exist, the function must approach the same value from both directions—crucial for piecewise and absolute value functions ($|x|/x$). **Key Learnings:** - If direct substitution leads to $0/0$, use **algebraic manipulation** (factoring or rationalizing) to find the "hidden" value. - Trigonometric limits often require rewriting $\tan x$ as $\frac{\sin x}{\cos x}$ or adjusting the argument to match the fundamental limit $\frac{\sin \theta}{\theta}$. - A denominator approaching zero while the numerator is non-zero results in a limit that **does not exist** ($\infty$).