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Here, we show you a step-by-step solved example of limits by rationalizing. This solution was automatically generated by our smart calculator:
lim β‘ x β 0 ( 5 + x β 5 x ) \lim_{x\to0}\left(\frac{\sqrt{5+x}-\sqrt{5}}{x}\right) x β 0 lim β ( x 5 + x β β 5 β β )
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Applying rationalisation
lim β‘ x β 0 ( 5 + x β 5 x 5 + x + 5 5 + x + 5 ) \lim_{x\to0}\left(\frac{\sqrt{5+x}-\sqrt{5}}{x}\frac{\sqrt{5+x}+\sqrt{5}}{\sqrt{5+x}+\sqrt{5}}\right) x β 0 lim β ( x 5 + x β β 5 β β 5 + x β + 5 β 5 + x β + 5 β β )
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Intermediate steps
Multiply and simplify the expression within the limit
lim β‘ x β 0 ( 5 + x β 5 x 5 + x + 5 5 + x + 5 ) \lim_{x\to0}\left(\frac{\sqrt{5+x}-\sqrt{5}}{x}\frac{\sqrt{5+x}+\sqrt{5}}{\sqrt{5+x}+\sqrt{5}}\right) x β 0 lim β ( x 5 + x β β 5 β β 5 + x β + 5 β 5 + x β + 5 β β )
Multiplying fractions 5 + x β 5 x Γ 5 + x + 5 5 + x + 5 \frac{\sqrt{5+x}-\sqrt{5}}{x} \times \frac{\sqrt{5+x}+\sqrt{5}}{\sqrt{5+x}+\sqrt{5}} x 5 + x β β 5 β β Γ 5 + x β + 5 β 5 + x β + 5 β β
lim β‘ x β 0 ( ( 5 + x β 5 ) ( 5 + x + 5 ) x ( 5 + x + 5 ) ) \lim_{x\to0}\left(\frac{\left(\sqrt{5+x}-\sqrt{5}\right)\left(\sqrt{5+x}+\sqrt{5}\right)}{x\left(\sqrt{5+x}+\sqrt{5}\right)}\right) x β 0 lim β ( x ( 5 + x β + 5 β ) ( 5 + x β β 5 β ) ( 5 + x β + 5 β ) β )
The first term (a a a 5 + x \sqrt{5+x} 5 + x β
The second term (b b b 5 \sqrt{5} 5 β
Solve the product of difference of squares ( 5 + x β 5 ) ( 5 + x + 5 ) \left(\sqrt{5+x}-\sqrt{5}\right)\left(\sqrt{5+x}+\sqrt{5}\right) ( 5 + x β β 5 β ) ( 5 + x β + 5 β )
lim β‘ x β 0 ( ( 5 + x ) 2 β ( 5 ) 2 x ( 5 + x + 5 ) ) \lim_{x\to0}\left(\frac{\left(\sqrt{5+x}\right)^2- \left(\sqrt{5}\right)^2}{x\left(\sqrt{5+x}+\sqrt{5}\right)}\right) x β 0 lim β ( x ( 5 + x β + 5 β ) ( 5 + x β ) 2 β ( 5 β ) 2 β )
Cancel exponents 1 2 \frac{1}{2} 2 1 β 2 2 2
lim β‘ x β 0 ( 5 + x β ( 5 ) 2 x ( 5 + x + 5 ) ) \lim_{x\to0}\left(\frac{5+x- \left(\sqrt{5}\right)^2}{x\left(\sqrt{5+x}+\sqrt{5}\right)}\right) x β 0 lim β ( x ( 5 + x β + 5 β ) 5 + x β ( 5 β ) 2 β )
Cancel exponents 1 2 \frac{1}{2} 2 1 β 2 2 2
lim β‘ x β 0 ( 5 + x β 5 x ( 5 + x + 5 ) ) \lim_{x\to0}\left(\frac{5+x-5}{x\left(\sqrt{5+x}+\sqrt{5}\right)}\right) x β 0 lim β ( x ( 5 + x β + 5 β ) 5 + x β 5 β )
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Multiply and simplify the expression within the limit
lim β‘ x β 0 ( 5 + x β 5 x ( 5 + x + 5 ) ) \lim_{x\to0}\left(\frac{5+x-5}{x\left(\sqrt{5+x}+\sqrt{5}\right)}\right) x β 0 lim β ( x ( 5 + x β + 5 β ) 5 + x β 5 β )
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Explain this step further
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Subtract the values 5 5 5 β 5 -5 β 5
lim β‘ x β 0 ( x x ( 5 + x + 5 ) ) \lim_{x\to0}\left(\frac{x}{x\left(\sqrt{5+x}+\sqrt{5}\right)}\right) x β 0 lim β ( x ( 5 + x β + 5 β ) x β )
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Simplify the fraction x x ( 5 + x + 5 ) \frac{x}{x\left(\sqrt{5+x}+\sqrt{5}\right)} x ( 5 + x β + 5 β ) x β x x x
lim β‘ x β 0 ( 1 5 + x + 5 ) \lim_{x\to0}\left(\frac{1}{\sqrt{5+x}+\sqrt{5}}\right) x β 0 lim β ( 5 + x β + 5 β 1 β )
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Intermediate steps
Evaluate the limit lim β‘ x β 0 ( 1 5 + x + 5 ) \lim_{x\to0}\left(\frac{1}{\sqrt{5+x}+\sqrt{5}}\right) x β 0 lim β ( 5 + x β + 5 β 1 β ) x x x 0 0 0
1 5 + 0 + 5 \frac{1}{\sqrt{5+0}+\sqrt{5}} 5 + 0 β + 5 β 1 β
Add the values 5 5 5 0 0 0
1 5 + 5 \frac{1}{\sqrt{5}+\sqrt{5}} 5 β + 5 β 1 β
Combining like terms 5 \sqrt{5} 5 β 5 \sqrt{5} 5 β
1 2 5 \frac{1}{2\sqrt{5}} 2 5 β 1 β
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Evaluate the limit lim β‘ x β 0 ( 1 5 + x + 5 ) \lim_{x\to0}\left(\frac{1}{\sqrt{5+x}+\sqrt{5}}\right) lim x β 0 β ( 5 + x β + 5 β 1 β ) x x x 0 0 0
1 2 5 \frac{1}{2\sqrt{5}} 2 5 β 1 β
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Explain this step further
ξ Final answer to the exercise
1 2 5 \frac{1}{2\sqrt{5}} 2 5 β 1 β ξ