Limit Questions

Solution : $\displaystyle{\lim_{x \to \infty}}$$x^2 + x + 1\over {3x^2 + 2x – 5}$ It is ($\infty\over \infty$ form),   Put x = $1\over y$ = $\displaystyle{\lim_{y \to 0}}$ $1 + y + y^2\over {3 + 2y – 5y^2}$ = $1\over 3$ Similar Questions Evaluate : $\displaystyle{\lim_{x \to 0}}$ $x^3 cotx\over {1-cosx}$ Evaluate : \(\displaystyle{\lim_{x \to […]

Solution : $\displaystyle{\lim_{x \to 0}}$ $x^3 cosx\over {sinx(1-cosx)}$ = $\displaystyle{\lim_{x \to 0}}$ $x^3 cosx(1 + cosx)\over {sinxsin^2x}$ = $\displaystyle{\lim_{x \to 0}}$ ${x^3\over sin^3x}.cosx(1 + cosx)$ = 2 Similar Questions Evaluate the limit : $\displaystyle{\lim_{x \to \infty}}$ $x^2 + x + 1\over {3x^2 + 2x – 5}$ Evaluate : $\displaystyle{\lim_{x \to \infty}}$ $({7x^2+1\over 5x^2-1})^{x^5\over {1-x^3}}$ Evaluate

Solution : Here f(x) = ${7x^2+1\over 5x^2-1}$ $\phi$(x) = ${x^5\over {1-x^3}}$ = $x^2x^3\over 1-x^3$ = $x^2\over {1\over x^3}-1$ $\therefore$ $\displaystyle{\lim_{x \to \infty}}$ f(x) = $7\over 5$ &  $\displaystyle{\lim_{x \to \infty}}$ $\phi$(x) $\rightarrow$ – $\infty$ $\implies$ $\displaystyle{\lim_{x \to \infty}}$ $(f(x))^{\phi (x)}$ = $({7\over 5})^{-\infty}$ = 0 Similar Questions Evaluate : $\displaystyle{\lim_{x \to 0}}$ $x^3 cotx\over {1-cosx}$

Solution : $\displaystyle{\lim_{x \to 0}}$ $xln(1+2tanx)\over 1-cosx$ = $\displaystyle{\lim_{x \to 0}}$ $xln(1+2tanx)\over {1-cosx\over x^2}.x^2$.$2tanx\over 2tanx$ = 4 Similar Questions Evaluate : $\displaystyle{\lim_{x \to 0}}$ $x^3 cotx\over {1-cosx}$ Evaluate : $\displaystyle{\lim_{x \to \infty}}$ $({7x^2+1\over 5x^2-1})^{x^5\over {1-x^3}}$ Evaluate the limit : $\displaystyle{\lim_{x \to \infty}}$ $x^2 + x + 1\over {3x^2 + 2x – 5}$ Evaluate : \(\displaystyle{\lim_{x

Solution : $\displaystyle{\lim_{x \to 0}}$ $2(sin(2+x)-sin2)+xsin(2+x)\over x$ = $\displaystyle{\lim_{x \to 0}}$($2.2.cos(2+{x\over 2})sin{x\over 2}\over x$ + sin(2+x)) = $\displaystyle{\lim_{x \to 0}}$$2cos(2+{x\over 2})sin{x\over 2}\over {x\over 2}$ + $\displaystyle{\lim_{x \to 0}}$sin(2+x) = 2cos2 + sin2 Similar Questions Evaluate : $\displaystyle{\lim_{x \to 0}}$ $x^3 cotx\over {1-cosx}$ Evaluate : $\displaystyle{\lim_{x \to \infty}}$ $({7x^2+1\over 5x^2-1})^{x^5\over {1-x^3}}$ Evaluate : $\displaystyle{\lim_{x \to 0}}$

Solution : $\displaystyle{\lim_{x \to \infty}}$(${x^3+1\over x^2+1}-(ax+b)$) = 2 $\implies$ $\displaystyle{\lim_{x \to \infty}}$$x^3(1-a)-bx^2-ax+(1-b)\over x^2+1$ = 2 $\implies$ $\displaystyle{\lim_{x \to \infty}}$$x(1-a)-b-{a\over x}+{(1-b)\over x^2}\over 1+{1\over x^2}$ = 2 $\implies$ 1 – a = 0, -b = 2 $\implies$ a = 1, b = -2 Similar Questions Evaluate : $\displaystyle{\lim_{x \to 0}}$ $x^3 cotx\over {1-cosx}$ Evaluate : \(\displaystyle{\lim_{x \to