Squeeze arguments and infinite behavior

Understand the idea

If a function is trapped between two functions with the same limiting value, it shares that limit. Infinite limits describe unbounded output and require sign analysis on each side of a zero denominator. Limits at infinity describe end behavior. For rational functions divide by the dominant power or compare degrees; a horizontal asymptote may be crossed. Standard trigonometric limits use radians.

\lim_{x\to0}\frac{\sin x}{x}=1,\qquad\lim_{x\to\infty}\frac{ax^n+\cdots}{bx^n+\cdots}=\frac ab

Worked example

Find $\displaystyle \lim_{x\to0}x^2\cos(1/x)$ and $\displaystyle \lim_{x\to\infty}(3x^2+1)/(2x^2-5)$ .

Since

\displaystyle -x^2\le x^2\cos(1/x)\le x^2

, the first limit is

\displaystyle 0

by squeezing. Dividing numerator and denominator of the second expression by

\displaystyle x^2

gives

\displaystyle (3+1/x^2)/(2-5/x^2)\to3/2

.

Try it yourself

Find the left and right limits of $\displaystyle 1/(x-2)$ at $\displaystyle 2$ .

Show solution and reasoning

The denominator approaches zero through negative values from the left and positive values from the right. The limits are $\displaystyle -\infty$ and $\displaystyle +\infty$ , respectively.