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Wize University Calculus 1 Textbook > Limits

Computing Limits by Getting a Common Denominator

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Computing Limits by Getting a Common Denominator

When limit problems contain two fractions being added or subtracted, the technique is usually to get a common denominator. Sometimes getting a common denominator is enough. Sometimes we need to factor as well or use another limit technique to cancel things out.

How to get a Common Denominator

Remember the basic strategy for getting a common denominator between fractions:

ab+cd=adbd+bcbd=ad+bcbd\displaystyle\frac{a}{b} + \frac{c}{d} = \frac{ad}{bd}+\frac{bc}{bd} = \frac{ad+bc}{bd}

Cross Multiplication

Alternatively, you can use the cross multiply technique and skip the middle step:

ab+cd=ad+bcbd\displaystyle\frac{a}{b} + \frac{c}{d} = \frac{ad+bc}{bd}


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Example: Getting a Common Denominator

Find the following limit

lim⁡x→414−1xx−4\displaystyle\lim_{x\rightarrow4}\frac{\frac{1}{4}-\frac{1}{x}}{x-4}



lim⁡x→414−1xx−4=lim⁡x→4x−44xx−4=lim⁡x→4x−44x(x−4)=lim⁡x→4x−44x(x−4)=lim⁡x→414x=14(4)=116\displaystyle\lim_{x\rightarrow4}\frac{\frac{1}{4}-\frac{1}{x}}{x-4}=\displaystyle\lim_{x\rightarrow4}\frac{\frac{x-4}{4x}}{x-4} \\ \text{} \\=\displaystyle\lim_{x\rightarrow4}\frac{x-4}{4x(x-4)}\\ \text{} \\=\displaystyle\lim_{x\rightarrow4}\frac{\cancel{x-4}}{4x\cancel{(x-4})}\\ \text{} \\=\displaystyle\lim_{x\rightarrow4}\frac{1}{4x}\\ \text{} \\=\frac{1}{4(4)}\\ \text{} \\=\boxed{\frac{1}{16}}















Find the following limit

lim⁡x→101x−120−xx−10\displaystyle \lim_{x\rightarrow 10}\frac{\frac{1}{x}-\frac{1}{20-x}}{x-10}