If the left hand limit and right hand limit of a function at a point and value of the function at that point is plus infinity,then do we say that the limit exist because both the left limit and right limits and value of function are equal.
Like in the function $\frac{2+\cos x}{x^3\sin x}$,left hand limit,right hand limit and value of the function at zero are all plus infinity.
I know that if left hand limit is minus infinity and right hand limit is plus infinity and vice versa ,then the limit does not exist.
Please help me.Thanks.
Best Answer
The value of $\frac{2+\cos x}{x^3\sin x}$ at $x = 0$ is NOT $+\infty$. Rather, it is undefined at $x = 0$. The left and right limits are $+\infty$, as this is a well-defined concept, but by convention, expressions such as $\frac{2+\cos x}{x^3\sin x}$ are considered to be functions from subsets of $\Bbb R$ into $\Bbb R$. But $\Bbb R$ contains no such number as $+\infty$.
However, your actual question does not require the function to be defined at $x = 0$. Limits are about the behavior of the function around the point, not at the point itself. So it doesn't matter that the function is not defined there.
Since there is no such real number as $+\infty$, it is not true that the left and right limits exist. But instead, they are said to "diverge to $+\infty$, or "increase without bound". Since they both diverge to $+\infty$, the two-sided limit does as well: $$\lim_{x\to 0} \frac{2+\cos x}{x^3\sin x} = +\infty$$