Evaluate the following: (i) ​lim (x^2 - 5x + 6)/(x^2 -4) [x ∈ 2] (ii) lim (3x^2 - x -10)/(x^2 -4) [x ∈ 2]

Question

Evaluate the following:

(i) \(\lim\limits_{x\to 2}\) \(\frac{x^2 - 5x + 6}{x^2 - 4}\)

lim (x^2 - 5x + 6)/(x^2 -4) [x ∈ 2]

(ii) \(\lim\limits_{x\to 2}\) \(\frac{3x^2 - x -10}{x^2 - 4}\)

lim (3x^2 - x -10)/(x^2 -4) [x ∈ 2]

(iii) \(\lim\limits_{x\to 3}\) \(\frac{x^4 - 81}{2x^2 -5x - 3}\)

lim (x^4 - 81)/(2x^2 -5x - 3) [x ∈ 3]

(iv) \(\lim\limits_{x\to 0}\)\(\frac{\sqrt{1+x}+ \sqrt{1-x}}{1+x}\)

lim (√(1+x )+ √(1-x))/(1+x) [x ∈ 0]

(v) \(\lim\limits_{x \to 1}\) \(\frac{x^3 - 1}{x-1}\)

lim (x^3 - 1)/(x - 1) [x ∈ 3]

(vi) \(\lim\limits_{x \to 1}\) \(\frac{sin 5x}{2x}\)

lim (sin 5x)/(2x) [x ∈ 1]

(vii) \(\lim\limits_{x \to 0}\) \(\frac{e^{3x}-1}{x}\)

lim (e^3x - 1)/(2x) [x ∈ 0]

Answer 1

 (i) \(\lim\limits_{x\to 2}\) \(\frac{x^2 - 5x + 6}{x^2 - 4}\)

 (ii) \(\lim\limits_{x\to 2}\) \(\frac{3x^2 - x -10}{x^2 - 4}\)

(iii) \(\lim\limits_{x\to 3}\) \(\frac{x^4 - 81}{2x^2 -5x - 3}\)

(iv) \(\lim\limits_{x\to 0}\)\(\frac{\sqrt{1+x}+ \sqrt{1-x}}{1+x}\)

\(\lim\limits_{x\to 0}\)\(\frac{\sqrt{1+0}+ \sqrt{1-0}}{1+0}\) = 2

(v) \(\lim\limits_{x\to 0}\) \(\frac{x^3- 1}{x-1}\) = 3(1)^3-1 = 3

 (vi) \(\lim\limits_{x \to 0}\) \(\frac{sin 5x}{2x}\) = \(\frac{5}{2}\)\(\lim\limits_{x \to 0}\) \(\frac{sin 5x}{5x}\) = \(\frac{5}{2}\)

 (vii) \(\lim\limits_{x \to 0}\) \(\frac{e^{3x}-1}{x}\) = 3

 \(\lim\limits_{x \to 0}\) \(\frac{e^{3x}-1}{3x}\) = 3