Limits of a function

• What is the limit expression for $\lim_{x \to 2} \frac{x^3 - 8}{x - 2}$?


  $\lim_{x \to 2} \frac{x^3 - 8}{x - 2} =$$12$
• What does the expression $\lim_{x \to \infty} \frac{3x^3 + 2x^2}{7x^3 + 9}$ represent?


  It represents the limit of the rational function as $x$ approaches infinity.
• How would you evaluate $\lim_{x \to 0} \frac{\sqrt{x}}{x}$?


  By applying L'Hôpital's rule, the limit evaluates to $\infty$.
• What is the limit expression for $\lim_{x \to 0} \frac{81x}{x^2 + 4x + 1 - x}$?


  $\lim_{x \to 0} \frac{81x}{x^2 + 4x + 1 - x} =$$81$
• What is the significance of the expression $\lim_{x \to 0} \frac{x^3 + 2x^2 - x}{x^2}$?


  It indicates the behavior of the function as $x$ approaches zero.
• What types of indeterminate forms can occur with limits involving exponential and logarithmic functions?


  Common indeterminate forms include $\frac{0}{0}$ and $\frac{\infty}{\infty}$.
• What is the limit expression for $\lim_{x \to 0} (1 + x)^{\frac{1}{x}}$?


  $e$
• How would you evaluate $\lim_{x \to 0} \frac{e^{-x^2}}{x^2}$?


  By applying L'Hôpital's rule, the limit evaluates to $\infty$.
• What does $\lim_{x \to \infty} \frac{3x^3 + 2x^2}{x^3}$ simplify to?


  It simplifies to $3$.
• What are the key concepts related to limits involving polynomials and radicals?

  • Evaluating limits at specific points
  • Understanding indeterminate forms
  • Applying L'Hôpital's rule
  • Simplifying expressions before taking limits
• What are the common types of indeterminate forms in calculus?


  • $\frac{0}{0}$
  • $\frac{\infty}{\infty}$
  • $\infty - \infty$
  • $0 \cdot \infty$
  • $\infty^0$
  • $1^{\infty}$