Lesson 6 of 10Large inputs and unbounded outputs
Distinguish a finite limit at infinity from an infinite limit near a finite input, using signs and dominant terms.
Try
Compare two experiments with f(x)=1/x. In the first, choose positive inputs 10, 100 and 1000. In the second, choose positive inputs 0.1, 0.01 and 0.001. One experiment makes the inputs large; the other makes the outputs large. Write the outputs before naming a limit.
Predict
Predict what changes if each input is negative. Does an output approaching zero have to be positive? Do the left and right sides near zero necessarily grow in the same direction?
Experiment
Use the table as a sign and scale check. Plot the points on two separate sketches rather than forcing very different ranges onto one misleading scale.
| Input x | 1/x | Input x | 1/x |
|---|
| 10 | 0.1 | 0.1 | 10 |
| 100 | 0.01 | 0.01 | 100 |
| −10 | −0.1 | −0.1 | −10 |
| −100 | −0.01 | −0.01 | −100 |
The table suggests the pattern; the reciprocal formula establishes it. For example, to make |1/x| smaller than 1/1000, it is enough to require |x| greater than 1000. To make 1/x exceed 1000, require a positive x smaller than 1/1000.
Understand
A limit at infinity describes eventual behavior as inputs become arbitrarily large in magnitude. For 1/x, the limit is zero both as x tends to positive infinity and as x tends to negative infinity. The line y=0 is a horizontal asymptote. Approaching zero from below still means approaching zero.
An infinite limit near a finite input describes outputs whose magnitude grows without bound. Near zero, 1/x approaches positive infinity from the right and negative infinity from the left. These are different one-sided behaviors. For 1/x², both sides approach positive infinity because the denominator is positive on either side. The line x=0 is a vertical asymptote. Infinity is not a real number assigned to f(0).
For rational end behavior, divide numerator and denominator by the highest relevant power of x. For (3x²−2x+4)/(x²+5), divide by x² to obtain (3−2/x+4/x²)/(1+5/x²). Reciprocal terms vanish as x grows in magnitude, and the remaining ratio approaches 3. This argument works at both ends. If degrees differ, analyze the remaining powers rather than blindly dividing leading coefficients: x²/x=x grows without bound at the positive end.
A horizontal asymptote is not a barrier. The function x/(x²+1) approaches zero at both ends and equals zero at x=0, where it crosses the asymptote. Asymptotes describe limiting behavior, not a prohibition on equality elsewhere.
Apply
In each practice item, state what is approaching what: is x approaching a finite number, positive infinity, or negative infinity? For a finite pole, examine the sign on each side. For end behavior, identify which terms vanish after division. Avoid entering infinity into ordinary arithmetic as though it were a measured value.
Compare growth with a ratio, not a finite plot
Both ln x and √x grow without bound, but that alone does not compare their scale. Their ratio tends to zero as x tends to infinity: L’Hôpital’s rule on the positive half-line gives (1/x)/(1/(2√x))=2/√x→0. Thus ln x is small relative to √x for sufficiently large x, even though neither function tends to zero. The derivative ratio also tends to zero, so their positive rates satisfy the same relative comparison. If L’Hôpital is unfamiliar, return after the limit-method lesson; infinity divided by infinity is an indeterminate form, not an answer.
For every fixed positive integer k, applying the same justified rule k times gives xᵏ/eˣ→0, since the last ratio is k!/eˣ. This compares fixed powers with exponentials; it does not permit a changing exponent k=x inside that argument. A finite graph window cannot establish the eventual statement for all larger inputs.
Transfer. Compare ln x with x^(1/3), and compare their rates, as x→∞. Then compare x² and e^(x/2). Explain why neither answer says the numerator itself approaches zero.
Review the asymptotic ratios
The first function ratio has derivative ratio 3/x^(1/3), which tends to zero. This also directly compares their rates: (1/x)/((1/3)x^(−2/3))=3/x^(1/3). For x²/e^(x/2), two differentiations give 8/e^(x/2)→0. All original numerators grow without bound; a ratio tending to zero is a relative-scale conclusion. State a positive domain and identify the indeterminate form before invoking the rule.
Continue
Next, a derivative is defined using a finite limit of slopes as an input increment tends to zero. Some graphs have vertical tangents, but an unbounded slope is not a finite derivative under the definition used here.
