Question

How would you solve the following limit:
limit (x -> 0+) (x-11)/sin(x)

Answer 1

you can do it manually, and figure out its negative infinity. but is there any algebraic way.

Answer 2

\[\lim_{x \rightarrow 0+} (x-11)/\sin(x)\]

Answer 3

\[\lim_{x \rightarrow 0}(sinx/x)=1\]

Answer 4

\[You know \lim_{x \rightarrow 0+} \frac{\sin x}{x}=1 \]

Answer 5

so: do the same trick I told you before: Divide the top and bottom by x

Answer 6

and \[\lim_{x \rightarrow 0}(11/sinx)\rightarrow \infty \]

Answer 7

\[\lim_{x \rightarrow 0+} \frac{\frac{11-x}{x}}{\frac{\sin x}{x}} = \lim_{x \rightarrow 0+} \frac{11/x-1}{\frac{\sin x}{x}}=\]\[\frac{\infty-1}{1}=\infty\]