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I came across the attached question in our calculus book. The limits in question are:

F(x), which approaches 0 from the left and 1 from the right as x goes to 2

J(x), which approaches 1 from the left and 0 from the right as x goes to 2.

Question B) ii. asks for the limit of [F(x) + J(x)] as x goes to 2. The left hand limits add up to 1, and the right hand limits do too, so the limit is 1 as x approaches 2 - the answer key matches this.

However, in C) ii., which asks for limit of [F(x)J(x)] as x approaches 2, the left hand limits multiply to 0 and the right hand limits multiply to 0, but the answer key has DNE. Am I missing something completely about how limits work? Or is the answer key wrong?

Thanks in advance! Edited photo of the page

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  • $\begingroup$ I agree that $\lim_\limits{x\to 2} f(x)j(x) = 0$ $\endgroup$ – Doug M Sep 30 '19 at 20:59
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By limit properties

$$\lim_{x\to 0^+} F(x)\cdot J(x)=\lim_{x\to 0^+} F(x)\cdot\lim_{x\to 0^+} J(x)=1\cdot0=0$$

$$\lim_{x\to 0^-} F(x)\cdot J(x)=\lim_{x\to 0^-} F(x)\cdot\lim_{x\to 0^-} J(x)=0\cdot1=0$$

therefore

$$\lim_{x\to 0} F(x)\cdot J(x)=0$$

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