Section 2.5 shows how to evaluate limits algebraically. I think that in some cases it is easier to solve algebraically, and it also saves some time rather than using the table method. However, I am not very good at math and usually make a lot of mistakes when trying to solve something algebraically so I will probably never use this method unless using a table is impossible.
Problems:
1. Solve this limit using algebra: lim (x^2 - 100) / (x-10) as x approaches 10.
I factored the numerator into (x+10)(x-10) / (x-10). Since (x-10) is present in both the numerator and the denominator, I cancelled it out, leaving me with the limit (x+10) as x approaches 10. Plugigng 10 in for x gave me the limit of (10+10) as x approaches 10 = 20.
2. Solve this limit: lim cot θ / csc θ as θ approaces 0.
Before I started to solve this problem, I changed cot and csc so that they were expressed in terms of sin and cos. Cot θ is equal to cos θ / sin θ and csc θ is equal to 1/sinθ. Since csc = 1/sinθ and is located on the denominator, I multiplied the entire limit by the reciprocal, sin θ / 1. That gave me: lim cosθ*sinθ / sinθ as x approaches 0. Because sinθ is present in both the numerator and denominator, you can cross it out. This leaves only lim cosθ as x approaches 0. If you plug in 0 for θ you get the lim cosθ as x approaches 0 = 1.
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