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 ${\lim }_{x\to c}f(x)$ exists if and only if ${\lim }_{x\to c^{-}}f(x)={\lim }_{x\to c^{+}}f(x)$.

• 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 ${\lim }_{x\to 0}\frac{\sin x}{x}=1$.

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Important Formulas

Below are the key formulas used in this exercise:

| Difference of Squares | $a^2-b^2=(a-b)(a+b)$ | | Trigonometric Identity | ${\lim }_{x\to 0}\frac{\sin x}{x}=1$ |

| Tangent Identity | ${\lim }_{x\to 0}\frac{\tan x}{x}=1$ |

| Limit of Sine (General) | ${\lim }_{x\to 0}\frac{\sin ax}{ax}=1$ |

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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 ($\infty$).