How to find and remove a discontinuity in $f(x) = \frac{x^2 - 4}{x - 2}$ ?

Factor the numerator: $f(x) = \frac{(x - 2)(x + 2)}{x - 2}$ . Cancel common factors: $f(x) = x + 2$ . Redefine $f(2) = 4$ .

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How to find and remove a discontinuity in $f(x) = \frac{x^2 - 4}{x - 2}$ ?

Factor the numerator: $f(x) = \frac{(x - 2)(x + 2)}{x - 2}$ . Cancel common factors: $f(x) = x + 2$ . Redefine $f(2) = 4$ .

How to ensure continuity of $f(x) = \begin{cases} x^2, & x \leq 1 \\ ax, & x > 1 \end{cases}$ ?

Set $x^2 = ax$ at $x = 1$ . Solve for $a$ : $1^2 = a(1)$ , so $a = 1$ .

How to determine if $f(x) = \frac{x^2 - 1}{x + 1}$ has a removable discontinuity?

Factor: $f(x) = \frac{(x - 1)(x + 1)}{x + 1}$ . Cancel: $f(x) = x - 1$ . Yes, at $x = -1$ .

How to redefine $f(x) = \frac{x^2 - 9}{x - 3}$ to be continuous?

Factor: $f(x) = \frac{(x - 3)(x + 3)}{x - 3}$ . Cancel: $f(x) = x + 3$ . Define $f(3) = 6$ .

How to find the value of 'a' to make $f(x) = \begin{cases} x + a, & x < 0 \\ x^2, & x \geq 0 \end{cases}$ continuous?

Set $x + a = x^2$ at $x = 0$ . Solve for $a$ : $0 + a = 0^2$ , so $a = 0$ .

Given $f(x) = \frac{x^2 - 4x + 3}{x - 1}$ , how do you find and remove the discontinuity?

Factor the numerator: $f(x) = \frac{(x - 1)(x - 3)}{x - 1}$ . Cancel the common factor: $f(x) = x - 3$ . Evaluate at x = 1: $f(1) = 1 - 3 = -2$ .

How do you ensure the function $f(x) = \begin{cases} cx^2, & x \leq 2 \\ 2x + c, & x > 2 \end{cases}$ is continuous?

Set the two pieces equal at x = 2: $c(2)^2 = 2(2) + c$ . Solve for c: $4c = 4 + c$ , so $3c = 4$ , and $c = \frac{4}{3}$ .

How do you find the limit of $f(x) = \frac{x^2 - 25}{x + 5}$ as x approaches -5?

Factor the numerator: $f(x) = \frac{(x - 5)(x + 5)}{x + 5}$ . Cancel the common factor: $f(x) = x - 5$ . Evaluate at x = -5: $f(-5) = -5 - 5 = -10$ .

How do you find the value of 'k' that makes $f(x) = \begin{cases} \frac{x^2 - 9}{x - 3}, & x \neq 3 \\ k, & x = 3 \end{cases}$ continuous?

Factor the numerator: $f(x) = \frac{(x - 3)(x + 3)}{x - 3}$ . Cancel the common factor: $f(x) = x + 3$ . Evaluate at x = 3: $f(3) = 3 + 3 = 6$ . Set k = 6.

How do you determine if the function $f(x) = \frac{x^2 + 3x + 2}{x + 2}$ has a removable discontinuity and remove it?

Factor the numerator: $f(x) = \frac{(x + 1)(x + 2)}{x + 2}$ . Cancel the common factor: $f(x) = x + 1$ . Yes, it has a removable discontinuity at x = -2. Evaluate at x = -2: $f(-2) = -2 + 1 = -1$ .

How do you test continuity for a piecewise function at x=a?

$\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)$

What is the general form of a rational function where removable discontinuities often occur?

$f(x) = \frac{p(x)}{q(x)}$ , where p(x) and q(x) are polynomials.

How do you find the value to 'fill the gap' at a removable discontinuity?

$f(a) = \lim_{x \to a} f(x)$ , where 'a' is the x-value of the discontinuity.

Write the simplified form of a function with a removable discontinuity at x=c after factoring.

If $f(x) = \frac{(x-c)g(x)}{(x-c)}$ , then the simplified function is $g(x)$ for $x \neq c$ .

What is the condition for the existence of a limit at a point?

$\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$

How can you express a piecewise function generally?

$f(x) = \begin{cases} f_1(x), & x \in D_1 \\ f_2(x), & x \in D_2 \\ ... \end{cases}$

How do you find the x-value(s) where a rational function might have discontinuities?

Solve $q(x) = 0$ where $f(x) = \frac{p(x)}{q(x)}$

How to redefine f(x) to remove discontinuity at x=a?

$f(x) = \begin{cases} original, & x \neq a \\ \lim_{x \to a} original, & x = a \end{cases}$

What is the simplified function after removing the discontinuity?

$f(x) = \frac{(x-a)g(x)}{(x-a)} = g(x)$

How do you solve for constant 'k' to make piecewise function continuous?

$\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = k$

What are the differences between removable and non-removable discontinuities?

Removable: Limit exists, can be 'fixed'. Non-removable: Limit doesn't exist, cannot be 'fixed'.

Compare jump and infinite discontinuities.

Jump: Function 'jumps' from one value to another. Infinite: Function approaches infinity (vertical asymptote).

What are the differences between direct substitution and factoring when evaluating limits?

Direct Substitution: Simple, works if function is continuous. Factoring: Used when direct substitution leads to indeterminate form.

Compare the graphs of continuous and discontinuous functions.

Continuous: No breaks, holes, or jumps. Discontinuous: Has breaks, holes, jumps, or asymptotes.

What are the differences between finding limits graphically and algebraically?

Graphically: Visual estimation of where the function is heading. Algebraically: Using techniques like factoring or rationalizing to find the exact value.

Compare the conditions for continuity at a point versus over an interval.

At a point: Limit exists and equals the function's value. Over an interval: Continuous at every point in the interval.

What are the differences between a hole and a vertical asymptote on a graph?

Hole: Removable discontinuity where the limit exists. Vertical Asymptote: Non-removable discontinuity where the function approaches infinity.

Compare the methods for finding discontinuities in rational functions versus piecewise functions.

Rational Functions: Look for zeros in the denominator. Piecewise Functions: Check the boundaries between the pieces.

What are the differences between the Intermediate Value Theorem and the Extreme Value Theorem?

Intermediate Value Theorem: Guarantees a function takes on every value between any two given points if continuous. Extreme Value Theorem: Guarantees the function has a max and min if continuous.

Compare direct substitution and L'Hopital's rule for evaluating limits.

Direct Substitution: First attempt, works if not indeterminate. L'Hopital's Rule: Used for indeterminate forms like 0/0 or ∞/∞, involves derivatives.