What does \(\lim_{x\to a^-}f(x)=L\) mean?

What condition on one-sided limits makes \(\lim_{x\to a}f(x)\) exist?

If \(\lim_{x\to2^-}f(x)=5\) and \(\lim_{x\to2^+}f(x)=5\), find \(\lim_{x\to2}f(x)\).

If \(\lim_{x\to1^-}f(x)=3\) and \(\lim_{x\to1^+}f(x)=7\), does \(\lim_{x\to1}f(x)\) exist?

Evaluate \(\lim_{x\to\infty}\frac{3x^2+1}{x^2-5}\).

Evaluate \(\lim_{x\to\infty}\frac{2x+4}{5x-1}\).

Find \(\lim_{x\to2^+}\frac1{x-2}\).

Find \(\lim_{x\to2^-}\frac1{x-2}\).

Evaluate \(\lim_{x\to\infty}\frac{4x^3-x}{2x^3+7}\).

Evaluate \(\lim_{x\to\infty}\frac{x}{x^2+1}\).

For \(f(x)=\begin{cases}x+1,&x<0\2x+1,&x>0\end{cases}\), find \(\lim_{x\to0^-}f(x)\), \(\lim_{x\to0^+}f(x)\), and the two-sided limit.

For \(f(x)=\begin{cases}1,&x<3\4,&x>3\end{cases}\), determine whether \(\lim_{x\to3}f(x)\) exists.

Evaluate \(\lim_{x\to-\infty}\frac{5x^2+x}{2x^2-3}\).

Evaluate \(\lim_{x\to\infty}\frac{7x-1}{x^2+4}\) and interpret the horizontal behavior.

Find \(\lim_{x\to1^+}\frac{2}{(x-1)^2}\) and \(\lim_{x\to1^-}\frac{2}{(x-1)^2}\).

Find \(\lim_{x\to0^+}\frac{1}{x^3}\) and \(\lim_{x\to0^-}\frac{1}{x^3}\).

For \(\frac{x^2+1}{x-4}\), classify the behavior as \(x\to4^+\) and \(x\to4^-\).

A student says \(\lim_{x\to\infty}\frac{3x^2+10}{x}=3\) by comparing coefficients. Diagnose the error and find the correct behavior.

Construct a simple piecewise function whose left-hand limit at \(0\) is \(-1\) and right-hand limit at \(0\) is \(1\), and state the two-sided limit.

Explain why \(\lim_{x\to a}f(x)=+\infty\) is not a finite limit, even though it describes limiting behavior.