UNDERSTANDING \( \frac{\infty}{\infty} \) USING RATE OF CHANGE
Imagine two cars travelling on parallel tracks and they end their journey at the same point. That end point is like the final value of both functions. Since both cars stop at the same place, we cannot tell which one was “ahead” just by looking at the final position. This is exactly what happens in an indeterminate form like \( \frac{\infty}{\infty} \).
So instead of comparing the final point, we compare the rate at which they are moving near the final point. The derivative represents this rate of change. It tells us how fast the function is changing near that point. By comparing the derivatives, we are really comparing the speeds of the two cars near the finish.
Look at the graph below for a while. Think of Car A following \( y = 3x + 7 \) and Car B following \( y = 2^{x} \). For large values of \( x \), which point is moving faster? Even though both functions eventually head towards infinity, we can clearly see that Car B has a much higher rate of change \( \left( \frac{dy}{dx} \right) \) for large \( x \). Key idea: compare the rate, not the final value.
Examples of \( \frac{\infty}{\infty} \)
1) Limit becomes \( 0 \)
\( \displaystyle \lim_{x\to\infty} \frac{x^{2}+1}{x^{5}+1} \)
Rates: \( f'(x) = 2x,\quad g'(x) = 5x^{4} \).
Ratio: \( \displaystyle \frac{2x}{5x^{4}} = \frac{2}{5x^{3}} \to 0 \).
The denominator grows much faster, so the limit is \( 0 \).
2) Limit becomes \( \infty \)
\( \displaystyle \lim_{x\to\infty} \frac{x^{5}+3}{x^{2}+1} \)
Rates: \( f'(x) = 5x^{4},\quad g'(x) = 2x \).
Ratio: \( \displaystyle \frac{5x^{4}}{2x} = \frac{5}{2} x^{3} \to \infty \).
Now the numerator grows much faster, so the limit is \( \infty \).
3) Limit becomes a finite number
\( \displaystyle \lim_{x\to\infty} \frac{3x^{3}+7}{6x^{3}-2} \)
Rates: \( f'(x) = 9x^{2},\quad g'(x) = 18x^{2} \).
Ratio: \( \displaystyle \frac{9x^{2}}{18x^{2}} = \frac{1}{2} \).
Here the rates are proportional, so the limit is the finite number \( \frac{1}{2} \).
Understanding \( \frac{0}{0} \) Using the Same Idea
Now imagine both cars are approaching the same point on the road, but this time that point is at height \( 0 \) on the \( y \)-axis. As \( x \to 0 \), both functions also move towards \( 0 \), so their quotient looks like \( \frac{0}{0} \). Once again, this common final value (here \( 0 \)) does not tell us what the limit of the quotient should be.
We use the same idea as before: compare the rates of change near that common \( x \)-value. L'Hôpital's Rule says we replace the original functions by their derivatives and compare those rates instead. If the derivative of the numerator grows faster, the quotient tends to \( \infty \); if the derivative of the denominator grows faster, the quotient tends to \( 0 \); and if the rates stay in balance, we get a finite limit.
1) Limit becomes \( 1 \)
\( \displaystyle \lim_{x\to 0} \frac{\sin x}{x} \)
Rates: derivative of \( \sin x \) is \( \cos x \), and derivative of \( x \) is \( 1 \).
At \( x = 0 \): \( \cos 0 = 1 \), so we compare
\( \displaystyle \frac{\cos 0}{1} = \frac{1}{1} = 1 \).
Therefore, the limit is \( 1 \).
2) Limit becomes \( 0 \)
\( \displaystyle \lim_{x\to 0} \frac{x^{2}}{\ln(1+x)} \)
As \( x \to 0 \): \( x^{2} \to 0 \) and \( \ln(1+x) \to 0 \), so the form is \( \frac{0}{0} \).
Rates: derivative of \( x^{2} \) is \( 2x \), and derivative of \( \ln(1+x) \) is
\( \displaystyle \frac{1}{1+x} \).
At \( x = 0 \): \( 2x \to 0 \) and \( \frac{1}{1+0} = 1 \).
So the ratio of rates is
\( \displaystyle \frac{2x}{\frac{1}{1+x}} \to \frac{0}{1} = 0 \).
Therefore, the limit is \( 0 \).
3) Limit becomes \( \infty \)
\( \displaystyle \lim_{x\to 0} \frac{\tan x}{x^{3}} \)
As \( x \to 0 \): \( \tan x \to 0 \) and \( x^{3} \to 0 \), so again we have \( \frac{0}{0} \).
Rates: derivative of \( \tan x \) is \( \sec^{2} x \), and derivative of \( x^{3} \) is \( 3x^{2} \).
At \( x = 0 \): \( \sec^{2} 0 = 1 \) and \( 3x^{2} \to 0 \).
So the ratio of rates is
\( \displaystyle \frac{\sec^{2} x}{3x^{2}} \to \frac{1}{3x^{2}} \), which grows without bound as \( x \to 0 \).
Therefore, the limit is \( \infty \).
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