which function has infinite discontinuity?

$ f(x) = \frac{x^2 - 49}{x - 7} $

$ f(x) = \frac{1}{2x - 9} $

$ f(x) = x^5 - x^3 $

$ f(x) = \begin{cases} 2 & \text{if } x

Question

which function has infinite discontinuity?

$ f(x) = \frac{x^2 - 49}{x - 7} $

$ f(x) = \frac{1}{2x - 9} $

$ f(x) = x^5 - x^3 $

$ f(x) = \begin{cases} 2 & \text{if } x < 0 \\ 3 & \text{if } x \ge 0 \end{cases} $

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Continuity of Rational Functions" ], "new_concepts": [ "Infinite Discontinuity", "Types of Discontinuity" ], "current_concepts": [ "Continuity of Rational Functions", "Infinite Discontinuity", "Types of Discontinuity" ] } ## Analyze the discontinuity of each function $$ \begin{aligned} &\text{Option 1: } f(x) = \frac{x^2-49}{x-7} = \frac{(x-7)(x+7)}{x-7} = x+7 \quad (x eq 7) \implies \text{Removable discontinuity at } x = 7\ &\text{Option 2: } f(x) = \frac{1}{2x-9} \implies \lim_{x \to 4.5^+} f(x) = \infty, \ \lim_{x \to 4.5^-} f(x) = -\infty \implies \text{Infinite discontinuity at } x = 4.5\ &\text{Option 3: } f(x) = x^5 - x^3 \implies \text{Polynomial function, continuous everywhere}\ &\text{Option 4: } f(x) = \begin{cases} 2 & \text{if } x < 0 \ 3 & \text{if } x \ge 0 \end{cases} \implies \lim_{x \to 0^-} f(x) = 2, \ \lim_{x \to 0^+} f(x) = 3 \implies \text{Jump discontinuity at } x = 0 \end{aligned} $$ ## Identify the function with infinite discontinuity $$ f(x) = \frac{1}{2x-9} $$

(A) $f(x) = \frac{x^2 - 49}{x - 7}$

(B) $f(x) = \frac{1}{2x - 9}$

(C) $f(x) = x^5 - x^3$

(D) $f(x) = \begin{cases} 2 & \text{if } x < 0 \ 3 & \text{if } x \ge 0 \end{cases}$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Calculus", "Continuity of Rational Functions" ] }

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Explanation:

Analyze the discontinuity of each function

Identify the function with infinite discontinuity

Answer: