Define them in English please

Here are the definitions and explanations:

• Definition of Continuity and Limits

  • Continuity of a function at a point means that the function is defined at that point, its limit exists at that point, and this limit is equal to the function's value at that point. Graphically, a continuous function can be drawn without lifting the pen.
  • The limit of a function describes the behavior of the function as the input variable approaches a certain value (finite or infinite).

• Limits of Polynomial Functions For a polynomial function $P(x) = a_n x^n + \dots + a_1 x + a_0$:

  • At a finite point $c$: The limit is simply the value of the function at that point. $$\lim_{x \to c} P(x) = P(c)$$
  • At infinity ($\pm \infty$): The limit is that of its highest degree term. $$\lim_{x \to \pm \infty} P(x) = \lim_{x \to \pm \infty} a_n x^n$$

• Limits of Linear Functions A linear function is a special case of a polynomial function of degree 1, $f(x) = ax+b$:

  • At a finite point $c$: $$\lim_{x \to c} (ax+b) = ac+b$$
  • At infinity ($\pm \infty$): $$\lim_{x \to \pm \infty} (ax+b) = \lim_{x \to \pm \infty} ax$$ The limit will be $\pm \infty$ depending on the sign of $a$ and the direction of infinity.

• Limits of an Irrational Function An irrational function contains a variable under a radical (for example, $\sqrt{x}$).

  • At a point $c$ where the function is defined and continuous, the limit is obtained by direct substitution.
  • If direct substitution leads to an indeterminate form (like $\frac{0}{0}$ or $\frac{\infty}{\infty}$) or if the point is a boundary of the domain, techniques like multiplying by the conjugate expression can be used to simplify the expression before evaluating the limit.

• L'Hôpital's Rule L'Hôpital's Rule is used to evaluate limits of indeterminate forms $\frac{0}{0}$ or $\frac{\infty}{\infty}$. If $\lim_{x \to c} f(x) = 0$ and $\lim_{x \to c} g(x) = 0$ (or if both tend to $\pm \infty$), then:

  $$\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$$

  provided that the limit of the quotient of the derivatives exists.

• Limits of Rational Functions A rational function is a quotient of two polynomials, $R(x) = \frac{P(x)}{Q(x)}$:

  • At a finite point $c$ where $Q(c) \neq 0$: $$\lim_{x \to c} \frac{P(x)}{Q(x)} = \frac{P(c)}{Q(c)}$$
  • At a finite point $c$ where $Q(c) = 0$ and $P(c) \neq 0$: The limit is $\pm \infty$, and the sign must be determined by studying the left and right-hand limits.
  • At a finite point $c$ where $Q(c) = 0$ and $P(c) = 0$: This is an indeterminate form $\frac{0}{0}$. One can factor $(x-c)$ from the numerator and denominator, or use L'Hôpital's Rule.
  • At infinity ($\pm \infty$): The limit is that of the ratio of the highest degree terms of the numerator and denominator. $$\lim_{x \to \pm \infty} \frac{P(x)}{Q(x)} = \lim_{x \to \pm \infty} \frac{a_n x^n}{b_m x^m}$$