Wize University Calculus 1 Textbook > Limits

Indeterminate Forms

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Indeterminate Forms

Sometimes evaluating limit of a complicated function using limit rules results in indeterminate terms such as


00,∞∞,∞−∞,00,1∞,0⋅∞\frac{0}{0},\frac{\infty}{\infty},\infty-\infty,0^0,1^{\infty},0\cdot\infty


For these cases we have to use limit techniques to find the limit of the function.


Rational functions

For limits of the rational functions in the form of 00,\frac{0}{0}, extract the factor on numerator and denominator which makes them zero (The problematic factor) and cancel it out.





Extra Practice

Limits at infinity: Indeterminate forms

Which of the following statements is true about the functionf(x)=7x72+x12+15x72−5x12f\left(x\right)=\frac{7x^{\frac{7}{2}}+x^{\frac{1}{2}}+1}{5x^{\frac{7}{2}}-5x^{\frac{1}{2}}}?

Limits: Indeterminate forms

Evaluate the limit lim⁡x→0 ∣4x−1∣−∣4x+1∣x\displaystyle \lim_{x\to0}\ \frac{\left|4x-1\right|-\left|4x+1\right|}{x}.

🦊 TRICKY! Evaluate the limit,lim⁡x→0∣2x−1∣−∣2x+1∣x\displaystyle \lim_{x\rightarrow0}\frac{\lvert 2x-1\rvert-\lvert 2x+1\rvert}{x} .

🦊 TRICKY! Evaluate lim⁡x→0sin⁡2x−3sin⁡xsin⁡x\displaystyle \lim_{x\rightarrow 0}\frac{\sin{2x}-3\sin{x}}{\sin{x}} . Hint: use trigonometric identitysin⁡(2x)=2sin⁡xcos⁡x\sin(2x)=2\sin x\cos x .

Evaluate the limit lim⁡x→0 ∣4x−1∣−∣4x+1∣x\lim_{x\to0}\ \frac{\left|4x-1\right|-\left|4x+1\right|}{x}.

Indeterminate Forms: One-sided limits

Evaluate:

lim⁡x→1−∣x−1∣x−1\lim_{x \rightarrow 1^-} \frac{|x-1|}{x-1}

Limits at infinity: Indeterminate forms

Which of the following statements is true about the functionf(x)=7x72+x12+15x72−5x12f\left(x\right)=\frac{7x^{\frac{7}{2}}+x^{\frac{1}{2}}+1}{5x^{\frac{7}{2}}-5x^{\frac{1}{2}}}?

Limits at infinity: Indeterminate forms

Which of the following statements is true about the functionf(x)=7x72+x12+15x72−5x12f\left(x\right)=\frac{7x^{\frac{7}{2}}+x^{\frac{1}{2}}+1}{5x^{\frac{7}{2}}-5x^{\frac{1}{2}}}?

Limits at infinity: Indeterminate forms

Which of the following statements is true about the functionf(x)=7x72+x12+15x72−5x12f\left(x\right)=\frac{7x^{\frac{7}{2}}+x^{\frac{1}{2}}+1}{5x^{\frac{7}{2}}-5x^{\frac{1}{2}}}?

Limits at infinity: Indeterminate forms

Which of the following statements is true about the functionf(x)=7x72+x12+15x72−5x12f\left(x\right)=\frac{7x^{\frac{7}{2}}+x^{\frac{1}{2}}+1}{5x^{\frac{7}{2}}-5x^{\frac{1}{2}}}?

Limits: Indeterminate forms

Evaluate the limit lim⁡x→0 ∣4x−1∣−∣4x+1∣x\displaystyle \lim_{x\to0}\ \frac{\left|4x-1\right|-\left|4x+1\right|}{x}.