Evaluate the following limits
\(\lim _{x \rightarrow \sqrt{2}} \frac{x^{4}-4}{x^{2}+3 \sqrt{2} x-8}\)

2

(a)

Evaluate the following limits

\(\lim_ { x\rightarrow 3 }{ lim } \frac { { x }^{ 4 }-81 }{ { 2x }^{ 2 }-5x-3 } \)

2

(a)

Find the derivative of x² – 2 at x = 10.

2

(a)

Find the derivative of 99x at x = 100.

2

(a)

Find the derivate of the following functions from first principle
x^2/3

2

(a)

Find the derivative of
(i) 2x-\(\frac{3}{4}\)
(ii) (5x³ + 3x - 1) (x - 1)
(iii) \(x^{-3}(5+3x)\)
(iv) x⁵(3-6x^-9)
(v) x^-4(3-4x^-5)
(vi) \(\frac{2}{x+1}-\frac{x^{2}}{3x-1}\)

2

(a)

Evaluate:\(\underset { x\rightarrow 0 }{ Lt } \left( 1+2x \right) ^{ \frac { 5 }{ x } }\)

2

(a)

Evaluate:\(\underset { x\rightarrow 0 }{ Lt } \frac { { a }^{ x }+{ b }^{ x }+{ c }^{ x }-3 }{ x } ;a,b,c>0\)

2

(a)

Evaluate:\(\underset { x\rightarrow 0 }{ Lt } \frac { log(a+x)-loga }{ x } ;a>0\)

2

(a)

If f is an even function, then prove that \(\overset{lim f(x)}{x\rightarrow 0^{-}}=\overset{lim f(x)}{x\rightarrow 0^{+}}\)

2

(a)

Evaluate \(\lim_{x\rightarrow 2} \frac{x^{3}-6x^{2}+11x-6}{x^{2}-6x+8}\)

2

(a)

Evaluate \(\lim_{x\rightarrow 0}\frac{x^{3}cot x}{1-cos x}\)

2

(a)

Evaluate \(\lim_{x\rightarrow \pi} \frac{1+cos^{3}x}{1-cos2x}\)

2

(a)

Find the derivative of \(\frac{3^{x}}{x+tan x}\) w.r.t.x.

2

(a)

Differentiate \(\sqrt{sin x}\) w.r.t. x by first principle method.

2

(a)