Evaluate the following: (i) lim (sin ax)/(sin bx), [x∈0] (ii) lim (sin^2 3x)/(x^2), [x∈0]

Question

Evaluate the following:

(i) \(\lim \limits_{x \to 0} \cfrac{sin \,ax}{sin \,bx}\)

 lim (sin ax)/(sin bx), [x∈0]

(ii) \(\lim \limits_{x \to 0}\)\(\cfrac{sin^2 3x}{x^2}\)

lim (sin^2 3x)/(x^2), [x∈0]

(iii) \(\lim \limits_{x \to 0}\) \(\cfrac{1-cos x}{x^2}\)

lim (1-cosx)/(x^2), [x∈0]

(iv) \(\lim \limits_{x \to \pi}\) \(\cfrac{sin(\pi -x)}{\pi(\pi - x)}\)

lim sin(π - x)/π(π - x), [x∈ π]

(v) \(\lim \limits_{x \to 0}\) \(\cfrac{ax + x\,cosx}{bsinx}\)

lim (ax + xcos x)/(bsinx), [x∈0]

(vi) \(\lim \limits_{x \to 0}\) \(\cfrac{cos\,2x - 1}{cos \,x -1}\)

lim (cos 2x -1)/(cos x -1), [x∈0]

(vii) \(\lim \limits_{x \to \frac{\pi}{2}}\) \(\cfrac{tan 2x}{x - {\frac{\pi}{2}}}\)

lim (tan2x)/(x -π/2), [x∈ π/2]

Answer 1

(i)

(ii)

= 1 × 3 × 1 × 3 = 9

(iii)

\(2 \times 1 \times \frac{1}{2} \times 1 \times \frac{1}{2} = \frac{1}{2}\)

iv) Put π – x = y, x → π, y → 0

(v) 

(vi) 

(vii) Put x –\(\frac{\pi}{2}\) = y, x → \(\frac{\pi}{2}\), y → 0