One-Sided Limits and Limits at Infinity | NIMCET

Quick answer: The left-hand limit $\lim_{x \to a^-}f(x)$ uses values just below a; the right-hand limit $\lim_{x \to a^+}f(x)$ uses values just above. $\lim_{x \to a}f(x)$ exists only if both are finite and equal. As $x \to \infty$, a rational function's limit depends on degrees: 0 if the numerator's degree is lower, the ratio of leading coefficients if equal, and ±∞ if higher. The sandwich theorem says that if $g \le f \le h$ and $g, h \to L$, then $f \to L$.

This continues from standard limits and L'Hospital's rule.

One-sided limits

Functions where one-sided limits matter

For $[x]$ at any integer n: LHL $= n - 1$ and RHL $= n$.

Limits at infinity

For $\frac{a_mx^m + \ldots}{b_nx^n + \ldots}$ as $x \to \infty$:

Method: divide the numerator and denominator by the highest power of x in the denominator.

Other useful results:

Sandwich (squeeze) theorem

If $g(x) \le f(x) \le h(x)$ near a, and $\lim g = \lim h = L$, then $\lim f = L$.

Example: since $-\lvert x\rvert \le x\sin\frac{1}{x} \le \lvert x\rvert$ and both bounds go to 0, $\lim_{x \to 0}x\sin\frac{1}{x} = 0$.

Worked examples

Example 1: a piecewise function

Let $f(x) = 2x + 1$ for $x < 1$ and $f(x) = kx^2$ for $x \ge 1$. Find k so that $\lim_{x \to 1}f(x)$ exists.

LHL $= 3$, RHL $= k$, so $k = 3$.

Example 2: rational function at infinity

$\lim_{x \to \infty}\frac{3x^2 - 5x + 1}{2x^2 + 7} = \frac{3}{2}$

Example 3: square roots at infinity

$\lim_{x \to \infty}\left(\sqrt{x^2 + x} - x\right) = \lim_{x \to \infty}\frac{x}{\sqrt{x^2 + x} + x} = \frac{1}{2}$

Example 4: greatest integer

Find $\lim_{x \to 0}\frac{[x]}{x}$ if it exists.

RHL: for small positive x, $[x] = 0$, so the value is 0. LHL: for small negative x, $[x] = -1$, so $\frac{-1}{x} \to +\infty$. The limit does not exist.

Example 5: sandwich with the greatest integer

Find $\lim_{x \to \infty}\frac{[x]}{x}$.

Since $x - 1 < [x] \le x$, we get $1 - \frac{1}{x} < \frac{[x]}{x} \le 1$, so the limit is 1.

Summary table

Exam tips

• Whenever $[x]$, $\lvert x\rvert$ or a piecewise rule appears, compute LHL and RHL separately.
• For $x \to -\infty$ with square roots, remember $\sqrt{x^2} = -x$.
• Bounded × (something going to 0) gives 0.

Next, read continuity and types of discontinuity, or see the NIMCET Calculus chapter.

Common mistakes

• Saying a limit exists when only one side is computed.
• Taking $[-0.5] = 0$ instead of −1.
• Writing $\sqrt{x^2} = x$ for negative x.
• Treating $\infty - \infty$ as 0.

Frequently Asked Questions

When does the limit of a function exist?

When the left-hand and right-hand limits both exist, are finite and are equal.

What is the limit of [x] as x approaches an integer n?

The left-hand limit is n − 1 and the right-hand limit is n, so the limit does not exist.

How do you find the limit of a rational function at infinity?

Compare degrees: 0 if the top degree is smaller, the ratio of leading coefficients if equal, and infinite if larger.

What is the sandwich theorem?

If g(x) ≤ f(x) ≤ h(x) and both g and h tend to L, then f also tends to L.

What is the limit of sin x / x as x approaches infinity?

It is 0, because sin x is bounded and 1/x goes to 0.