Combining techniques [Example] [Algebraic limits]

Sometimes, multiple techniques are necessary to evaluate a limit algebraically. Here is a question that uses the techniques of combining fractions and multiplying by the conjugate.
I make some simplifications when directly substituting 𝑟 = 0 into the equation. In reality, infinity is not a number. I treat infinity as a “number” (0:34), or rather, the answer to division by zero because I personally need a placeholder to recognize that the fraction becomes really large in magnitude. If I do not use that infinity placeholder, it will be difficult for me to get a sense of how the equation behaves and thereby recognizing the indeterminate form “∞ − ∞”.
Another simplification is that I do not consider the one-sided limits. The question states “𝑟 → 0”, which implies that we should examine both 𝑟 → 0⁻ (approach 0 from the left) and 𝑟 → 0⁺ (approach 0 from the right). Let us do that now.
The limit 𝑟 → 0⁻ means that 𝑟 is a small number close to zero but is negative, i.e. to the left of 𝑟 = 0. Dividing by a small, negative number yields a large, negative number. Using the ∞ placeholder, the equation looks like (−∞) − (−∞) = −∞ + ∞, which is the indeterminate form “∞ − ∞”. Notice the minus sign between the two fractions. Subtracting a large, negative number is equivalent to adding a large, positive number.
The right limit, 𝑟 → 0⁺, means that 𝑟 is a small, positive number. Dividing by a small, positive number gives a large, positive number. The equation looks like (+∞) − (+∞) = ∞ − ∞, which is the same indeterminate form as the left limit.
The fractions in the equation individually approach different infinities, positive or negative, whether 𝑟 approaches 0 from the left or from the right. In either of the one-sided limits, we have the same indeterminate form “∞ − ∞”.
ᴄᴏʀʀɪɢᴇɴᴅᴜᴍ
The title card in 0:01 has the wrong equation. There does not seem to be a way to update the video after uploading. Thank you for your understanding!
Original content © 2023 Jung-Lynn Jonathan Yang
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[@BekzatJumabekov-t2t]

Good job 👌👍

[@surendrakverma555]

Good 👍

[@glorymanheretosleep]

Do more of these examples with trig functions too.