determine whether the following function is continuous at a. use the continuity checklist to justify your answer.

\y = \frac{4x - 5}{x^2 - 9x + 20}, a = 5\

select all that apply.

a. the function is continuous at $a = 5$.
b. the function is not continuous at $a = 5$ because $f(5)$ is undefined.
c. the function is not continuous at $a = 5$ because $\lim_{x \to 5} f(x)$ does not exist.
d. the function is not continuous at $a = 5$ because $\lim_{x \to 5} f(x) \
eq f(5)$.

Question

determine whether the following function is continuous at a. use the continuity checklist to justify your answer.

\y = \frac{4x - 5}{x^2 - 9x + 20}, a = 5\

select all that apply.

a. the function is continuous at $a = 5$.
b. the function is not continuous at $a = 5$ because $f(5)$ is undefined.
c. the function is not continuous at $a = 5$ because $\lim_{x \to 5} f(x)$ does not exist.
d. the function is not continuous at $a = 5$ because $\lim_{x \to 5} f(x) \
eq f(5)$.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Continuity Checklist" ], "new_concepts": [], "current_concepts": [ "Continuity Checklist", "Rational Function Continuity" ] } ## Evaluate the function at the given point $$ \begin{aligned} &f(x) = \frac{4x - 5}{x^2 - 9x + 20}\ &f(5) = \frac{4(5) - 5}{5^2 - 9(5) + 20} = \frac{15}{25 - 45 + 20} = \frac{15}{0} \implies \text{undefined} \end{aligned} $$ ## Evaluate the limit of the function as x approaches 5 $$ \begin{aligned} &\lim_{x \to 5} \frac{4x - 5}{(x-4)(x-5)}\ &\lim_{x \to 5^+} \frac{4x - 5}{(x-4)(x-5)} = \frac{15}{(1)(0^+)} = +\infty\ &\lim_{x \to 5^-} \frac{4x - 5}{(x-4)(x-5)} = \frac{15}{(1)(0^-)} = -\infty\ &\lim_{x \to 5} f(x) \text{ does not exist} \end{aligned} $$ ## Apply the continuity checklist $$ \begin{aligned} &1.\ f(5) \text{ is undefined} \implies \text{Not continuous}\ &2.\ \lim_{x \to 5} f(x) \text{ does not exist} \implies \text{Not continuous}\ &3.\ \lim_{x \to 5} f(x) e f(5) \implies \text{Not continuous} \end{aligned} $$

A. The function is continuous at a = 5.

B. The function is not continuous at a = 5 because f(5) is undefined.

C. The function is not continuous at a = 5 because $\lim_{x \to 5} f(x)$ does not exist.

D. The function is not continuous at a = 5 because $\lim_{x \to 5} f(x) eq f(5)$.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Calculus", "Continuity Checklist" ] }

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QUESTION IMAGE

Question

Explanation:

Evaluate the function at the given point

Evaluate the limit of the function as x approaches 5

Apply the continuity checklist

Answer: