Use numerical and graphical evidence to guess the value of the limit $$\underset{x \to 1} \lim \frac{x^3-1}{\sqrt{x}-1}$$
I am a Calculus 1 student, and I'm not sure what this problem in my textbook means when it says, "use numerical and graphical evidence". I worked around with it a bit and found that the answer is 6. However, I'm not sure if this is what my professor wants.
The textbook is "Calculus: Early Transcendentals", 8th Edition, by James Stewart. This is problem 55a in section 2.2.
What does this question mean by, "Use numerical and graphical evidence"?
$$\underset{x \to 1} \lim \frac{x^3-1}{\sqrt{x}-1}$$
$$=\underset{x \to 1} \lim \frac{x^3-1}{\sqrt{x}-1}\cdot \frac{\sqrt{x} + 1}{\sqrt{x} + 1}$$
$$=\underset{x \to 1} \lim \frac{(x^3-1)(\sqrt{x} + 1)}{x-1}$$
$$=\underset{x \to 1} \lim \frac{(x-1)(x^2+x+1)(\sqrt{x} + 1)}{x-1}$$
$$=\underset{x \to 1} \lim \;\ (x^2+x+1)(\sqrt{x} + 1)$$
$$= ((1)^2+(1)+1)(\sqrt{1}+1)=(3)(2)=6$$
Best Answer
To guess the limit, see this graph: