determine the following limits, using $\infty$ or $-\infty$ when appropriate, or state that they do not exist.
a. $\lim_{x \to 1^+} \frac{x^2 - 8x + 15}{(x - 1)^2}$
b. $\lim_{x \to 1^-} \frac{x^2 - 8x + 15}{(x - 1)^2}$
c. $\lim_{x \to 1} \frac{x^2 - 8x + 15}{(x - 1)^2}$

a. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. $\lim_{x \to 1^+} \frac{x^2 - 8x + 15}{(x - 1)^2} = $ (simplify your answer.)
b. the limit does not exist and is neither $\infty$ nor $-\infty$.

b. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. $\lim_{x \to 1^-} \frac{x^2 - 8x + 15}{(x - 1)^2} = $ (simplify your answer.)
b. the limit does not exist and is neither $\infty$ nor $-\infty$.

c. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. $\lim_{x \to 1} \frac{x^2 - 8x + 15}{(x - 1)^2} = $ (simplify your answer.)
b. the limit does not exist and is neither $\infty$ nor $-\infty$.

Question

determine the following limits, using $\infty$ or $-\infty$ when appropriate, or state that they do not exist.
a. $\lim_{x \to 1^+} \frac{x^2 - 8x + 15}{(x - 1)^2}$
b. $\lim_{x \to 1^-} \frac{x^2 - 8x + 15}{(x - 1)^2}$
c. $\lim_{x \to 1} \frac{x^2 - 8x + 15}{(x - 1)^2}$

a. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. $\lim_{x \to 1^+} \frac{x^2 - 8x + 15}{(x - 1)^2} = $   (simplify your answer.)
b. the limit does not exist and is neither $\infty$ nor $-\infty$.

b. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. $\lim_{x \to 1^-} \frac{x^2 - 8x + 15}{(x - 1)^2} = $   (simplify your answer.)
b. the limit does not exist and is neither $\infty$ nor $-\infty$.

c. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. $\lim_{x \to 1} \frac{x^2 - 8x + 15}{(x - 1)^2} = $   (simplify your answer.)
b. the limit does not exist and is neither $\infty$ nor $-\infty$.

Accepted Answer

{ "quality": "clear", "question_count": 3, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Limits of Rational Functions", "One-Sided Limits", "Infinite Limits" ], "new_concepts": [], "current_concepts": [ "Limits of Rational Functions", "One-Sided Limits", "Infinite Limits" ] } ## Evaluate the limit of the numerator and denominator as $x \to 1$ $$ \lim_{x \to 1} (x^2 - 8x + 15) = 1^2 - 8(1) + 15 = 8 > 0 $$ $$ \lim_{x \to 1} (x - 1)^2 = 0 $$ ## Analyze the sign of the function near $x = 1$ For all $x eq 1$, $(x - 1)^2 > 0$. Since the numerator is positive ($8 > 0$) and the denominator is positive and approaches $0$ as $x \to 1^+$ and $x \to 1^-$: $$ \lim_{x \to 1^+} \frac{x^2 - 8x + 15}{(x - 1)^2} = \infty $$ $$ \lim_{x \to 1^-} \frac{x^2 - 8x + 15}{(x - 1)^2} = \infty $$ ## Determine the two-sided limit Since both one-sided limits are $\infty$: $$ \lim_{x \to 1} \frac{x^2 - 8x + 15}{(x - 1)^2} = \infty $$ ## Question a A. $\lim_{x \to 1^+} \frac{x^2 - 8x + 15}{(x - 1)^2} = \infty$ B. The limit does not exist and is neither $\infty$ nor $-\infty$. ## Question b A. $\lim_{x \to 1^-} \frac{x^2 - 8x + 15}{(x - 1)^2} = \infty$ B. The limit does not exist and is neither $\infty$ nor $-\infty$. ## Question c A. $\lim_{x \to 1} \frac{x^2 - 8x + 15}{(x - 1)^2} = \infty$

B. The limit does not exist and is neither $\infty$ nor $-\infty$.

{ "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Calculus", "Infinite Limits" ] }

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QUESTION IMAGE

Question

Explanation:

Evaluate the limit of the numerator and denominator as \(x \to 1\)

Analyze the sign of the function near \(x = 1\)

Determine the two-sided limit

Answer:

Question a

Question b

Question c