Exponential and logarithmic limits

Evaluation of trigonometric limits: In order to evaluate trigonometric limits the following results are very useful:

⇒ sinx/x = 1

⇒ tanx/x = 1

⇒ sin⁻¹x/x = 1

⇒ tan⁻¹x/x = 1

⇒ sinx⁰/x = π/180

⇒ sin (x - a)/(x - a) = 1

⇒ tan (x - a)/(x - a) = 1

⇒ |sinx|/x does not exist

⇒ |tanx|/x does not exist

⇒ |sin (x - a)|/(x - a) does not exist

⇒ |tan (x - a)|/(x - a) does not exist.

Expansions useful in evaluation of limits:

1. (1 + x)ⁿ = 1 + nx + n [(n - 1)/2!] x² + [n (n - 1) (n - 2)/3!] x³ + …

2. eˣ = 1 + x/1! + x²/2! + x³/3!

3. aˣ = 1 + x/ 1! [logₑᵃ] + x²/2! [x²(logₑᵃ)²] + …

4. log (1 + x) = x - x²/2 + x³/3 - x⁴/4 + …

5. log (1 - x) = - x - x²/2 - x³/3 - x⁴/4 + …

6. sinx = x + x³/3! + x⁵/5! + …

7. cos x = 1 - x²/2! + x⁴/4! + …

8. tan x = x + x³/3 + 2/15 x⁵ + …

9. sin⁻¹ x = x + ½ x³/3 + ½ ¾ x⁵/5 + …

10. tan⁻¹ x = x - x³/3 + x⁵/5 + …

11. sec⁻¹ x = 1 + x²/2! + 5 x⁴/4! + …

Evaluate: (tan² x – 2 tanx - 3)/ (tan² x – 4 tanx + 3)

Solution: we have,

(tan x - 2 tanx - 3)/ (tan² x - 4 tanx + 3)

= (tanx - 3) (tanx + 1)/ (tanx - 3) (tanx - 1)

= (tanx + 1)/ (tanx - 1) = (3 + 1)/ (3 - 1) = 2

Evaluation of exponential and logarithm limits:

1. (aˣ - 1)/ x = logₑa, a > 0

2. logₑ (1 + x)/x = 1

3. (eˣ - 1)/x = logₑ = 1

4. logₐ (1 + x)/x = logₐe

Evaluate: 

Solution:We have,

= - e x - ½ = e/2