# Limits Examples

Here you will learn some limits examples for better understanding of limit concepts.

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

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

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

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.
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