student_001
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1. Let x0∈[0,1]. What is lim x→x0 f(x)?
2. Is f(x) continuous in [0,1]?
Limits and continuity
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1. Let x0∈[0,1]. What is lim x→x0 f(x)?
2. Is f(x) continuous in [0,1]?
Dr.Peterson
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What are your thoughts?
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I would have expected question 2 to come first.
HallsofIvy
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The only thing that could cause a problem is that denominator [imath]x^2+ x- 6[/imath]
Completing the square, [imath]x^2+ x- 6= x^2+ x+ 1/4- 1/4- 6= (x+ 1/2)^2- 25/4[/imath].
That will be 0 when [imath](x+ 1/2)^2= 25/4[/imath], [imath]x+ 1/2= \pm 5/2[/imath] so when x= -3 or x= 2. Neither of those is in [0, 1]. What does that tell you about the continuity of
[math]\frac{sin(e^x)}{x^2+ x- 6}[/math]
and now [imath]\lim_{x\to x_0} f(x)[/imath] is trivial!
Completing the square, [imath]x^2+ x- 6= x^2+ x+ 1/4- 1/4- 6= (x+ 1/2)^2- 25/4[/imath].
That will be 0 when [imath](x+ 1/2)^2= 25/4[/imath], [imath]x+ 1/2= \pm 5/2[/imath] so when x= -3 or x= 2. Neither of those is in [0, 1]. What does that tell you about the continuity of
[math]\frac{sin(e^x)}{x^2+ x- 6}[/math]
and now [imath]\lim_{x\to x_0} f(x)[/imath] is trivial!