Solved Calculus Limits: Rationalization and Derivative

Key takeaway: Both parts reward the same habit: look for a standard algebraic move first, then compare what remains with a familiar calculus pattern. One is a rationalization limit; the other is literally a derivative definition in disguise.

(d) $\lim_{t\to 0}\frac{\sqrt{t^2+9}-3}{t^2}$

This is a classic rationalization problem. Multiply top and bottom by the conjugate:

$\frac{\sqrt{t^2+9}-3}{t^2}\cdot\frac{\sqrt{t^2+9}+3}{\sqrt{t^2+9}+3}$

Then the numerator becomes

$\left(\sqrt{t^2+9}-3\right)\left(\sqrt{t^2+9}+3\right)=t^2+9-9=t^2$

So the expression simplifies to

$\frac{t^2}{t^2\left(\sqrt{t^2+9}+3\right)}=\frac{1}{\sqrt{t^2+9}+3}$

Now take the limit:

$\lim_{t\to 0}\frac{1}{\sqrt{t^2+9}+3}=\frac{1}{3+3}=\frac16$

(e) $\lim_{h\to 0}\frac{(3+h)^{-1}-3^{-1}}{h}$

Rewrite the powers as fractions:

$\frac{\frac{1}{3+h}-\frac13}{h}$

Combine the numerator:

$\frac{\frac{3-(3+h)}{3(3+h)}}{h}=\frac{\frac{-h}{3(3+h)}}{h}$

Cancel $h$:

$-\frac{1}{3(3+h)}$

Now let $h\to 0$:

$-\frac{1}{3\cdot 3}=-\frac19$

Final answers

• (d) $\frac16$
• (e) $-\frac19$

Part (e) is also the derivative of $f(x)=x^{-1}$ at $x=3$.

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Pitfalls the pros know 👇 For the radical limit, the main trap is stopping after rationalizing but forgetting that the $t^2$ in the denominator cancels completely. If you leave a zero in the denominator, something went wrong algebraically. For the difference quotient, don’t treat it as a raw limit of fractions before simplifying; the subtraction in the numerator has to be combined carefully before the $h$ cancels.

What if the problem changes? If the radical expression were $\lim_{t\to 0}\frac{\sqrt{t^2+9}-3}{|t|}$ instead of dividing by $t^2$, the behavior would be different because the denominator would shrink more slowly. For part (e), if the base point changed from $3$ to another number $a>0$, the same pattern would give

$\lim_{h\to 0}\frac{(a+h)^{-1}-a^{-1}}{h}=-\frac{1}{a^2}$

which is the derivative of $x^{-1}$ at $x=a$.

Tags: conjugate rationalization, difference quotient, derivative from first principles