Limits and continuity

[student_001]

1. Let x0∈[0,1]. What is lim x→x0 f(x)?

2. Is f(x) continuous in [0,1]?

 

[Dr.Peterson]

View attachment 28854
1. Let x0∈[0,1]. What is lim x→x0 f(x)?

2. Is f(x) continuous in [0,1]?

What are your thoughts?

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I would have expected question 2 to come first.

 

[HallsofIvy]

The only thing that could cause a problem is that denominator $x^2+ x- 6$

Completing the square, $x^2+ x- 6= x^2+ x+ 1/4- 1/4- 6= (x+ 1/2)^2- 25/4$.
That will be 0 when $(x+ 1/2)^2= 25/4$, $x+ 1/2= \pm 5/2$ so when x= -3 or x= 2. Neither of those is in [0, 1]. What does that tell you about the continuity of
$$\frac{sin(e^x)}{x^2+ x- 6}$$
and now $\lim_{x\to x_0} f(x)$ is trivial!