Limits Examples

Example 1 : If $\displaystyle{\lim_{x \to \infty}}$(${x^3+1\over x^2+1}-(ax+b)$) = 2, then find the value of a and b.

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

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Example 2 : Evaluate : $\displaystyle{\lim_{x \to 0}}$ $(2+x)sin(2+x)-2sin2\over x$

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

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Example 3 : Evaluate : $\displaystyle{\lim_{x \to 0}}$ $xln(1+2tanx)\over 1-cosx$

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

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Example 4 : Evaluate : $\displaystyle{\lim_{x \to \infty}}$ $({7x^2+1\over 5x^2-1})^{x^5\over {1-x^3}}$

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

Practice these given limits examples to test your knowledge on concepts of limits.