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Limits Worksheets - Download free PDFs

Limits

Basic

Substitution

1.\:\:\lim _{x\to 0}(\frac{1}{2})

2.\:\:\lim _{x\to 1}(2x^2-3x+5)

3.\:\:\lim _{x\to 2}(x(x-3))

4.\:\:\lim _{x\to 3}(\frac{3-x}{x^2+2x})

5.\:\:\lim _{x\to -1}(\frac{x+1}{x-1})^2

6.\:\:\lim _{x\to 5}(\frac{x^2-6x+8}{x-4})

7.\:\:\lim _{x\to -3}(\frac{(x-6)}{(5x^2-6x+8)})

8.\:\:\lim _{x\to 0}(\sqrt{x})

9.\:\:\lim _{x\to 3}(\frac{5x^{2}-8x-13}{x^{2}-5})

10.\:\:\lim _{x\to -7}(2x+5)

Limits

Basic

Infinite Limits

1.\:\:\lim _{x\to 0}(\frac{3}{5x^4})

2.\:\:\lim _{x\to 0}(\frac{2}{3x^2})

3.\:\:\lim _{x\to 0}(\ln (x))

4.\:\:\lim _{x\to 3}(\frac{x^3-5}{(x-3)^2})

5.\:\:\lim _{x\to 0+}(\frac{1}{x})

6.\:\:\lim _{x\to 0-}(\frac{1}{x})

7.\:\:\lim _{x\to 2-}(\frac{1}{x-2})

8.\:\:\lim _{x\to 2+}(\frac{1}{x-2})

9.\:\:\lim _{x\to 3+}(\frac{x+1}{x-3})

10.\:\:\lim _{x\to 5-}(\frac{2}{x-5})

Limits

Basic

Divergence

1.\:\:\lim _{x\to 0}(\frac{1}{x})

2.\:\:\lim _{x\to 5}(\frac{10}{x-5})

3.\:\:\lim _{x\to 1}(\frac{x}{x-1})

4.\:\:\lim _{x\to -2}(\frac{1}{x+2})

5.\:\:\lim _{x\to 5}(\frac{x}{x^2-25})

6.\:\:\lim _{x\to 2}\frac{|x-2|}{x-2}

7.\:\:\lim _{x\to 8}(\frac{1}{(x-8)})

8.\:\:\lim _{x\to 0}(\frac{1}{x^3})

9.\:\:\lim _{x\to 4}(\frac{1}{x-4})

10.\:\:\lim _{x\to 0}(\frac{|sin(x)|}{sin(x)})

Limits

Moderate

Limits at Infinity

1.\:\:\lim _{x\to \infty}(\frac{1}{x})

2.\:\:\lim _{x\to -\infty}(\frac{1}{x})

3.\:\:\lim _{x\to \infty}(x^2)

4.\:\:\lim _{x\to \infty}(x^3-x)

5.\:\:\lim _{x\to \infty}(5x^2+4x)

6.\:\:\lim _{x\to \infty}(2x^4-x^2-8x)

7.\:\:\lim _{x\to -\infty}(x^4+x^5)

8.\:\:\lim _{x\to \infty}(\frac{1}{x^{2}})

9.\:\:\lim _{x\to \infty}(e^x)

10.\:\:\lim _{x\to -\infty}(e^x)

Limits

Moderate

Factoring

1.\:\:\lim _{x\to 4}(\frac{x^2-6x+8}{x-4})

2.\:\:\lim _{x\to 2}(\frac{x^2-4}{x-2})

3.\:\:\lim _{x\to -2}(\frac{x+2}{x^3+8})

4.\:\:\lim _{x\to -5}(\frac{x^2-25}{x^2+4x-5})

5.\:\:\lim _{x\to 1}(\frac{x^3-x}{x^2-1})

6.\:\:\lim _{x\to -1}(\frac{x^2+x}{x+1})

7.\:\:\lim _{x\to 4}(\frac{(5x-20)}{(x-4)})

8.\:\:\lim _{x\to 5}(\frac{x^2-5x}{x-5})

9.\:\:\lim _{x\to -1}(\frac{x^3+x^2}{x+1})

10.\:\:\lim _{x\to 1}(\frac{x^2-1}{x-1})

Limits

Moderate

Square Roots

1.\:\:\lim _{x\to 4}(\frac{\sqrt{x}-2}{x-4})

2.\:\:\lim _{x\to 13}(\frac{\sqrt{x-4}-3}{x-13})

3.\:\:\lim _{x\to 0}(\frac{\sqrt{1+x}-1}{x})

4.\:\:\lim _{x\to 9}(\frac{\sqrt{x}-3}{x-9})

5.\:\:\lim _{x\to 4}(\frac{2-\sqrt{x}}{4-x})

6.\:\:\lim _{x\to \infty}(\sqrt{x^2+4}-x)

7.\:\:\lim _{t\to \infty}(\frac{\sqrt{t}+t^2}{6t-t^2})

8.\:\:\lim _{x\to \infty}(\frac{\sqrt{x}+x^2}{4x-x^2})

9.\:\:\lim _{x\to \infty}(\frac{\sqrt{x}}{x-5})

10.\:\:\lim _{x\to \infty}(\frac{\sqrt{x}+x^2}{3x-x^2})

Limits

Advanced

L'Hopital Rule

1.\:\:\lim _{x\to \infty}(\frac{\ln(x)}{x})

2.\:\:\lim _{x\to -\infty}(xe^x)

3.\:\:\lim _{x\to 0}(\frac{sin(x)}{x})

4.\:\:\lim _{x\to 0}(\frac{\tan(x)}{2x})

5.\:\:\lim _{x\to 0}(\frac{1-\sec^{2}(7x)}{(3x)^{2}})

6.\:\:\lim _{x\to 0}(x\ln (x))

7.\:\:\lim _{x\to 0}(\frac{\sin(5x)}{\sin(3x)})

8.\:\:\lim _{x\to \infty}(\frac{e^x}{x^2})

9.\:\:\lim _{x\to 0}(\frac{\sin(5x)}{2x})

10.\:\:\lim _{x\to 0}(\frac{1-\cos(2x)}{x})

Limits

Advanced

Chain Rule

1.\:\:\lim _{x\to \infty}e^{2-4x-8x^2}

2.\:\:\lim _{x\to \infty}(x^{\frac{1}{x}})

3.\:\:\lim _{x\to \infty}(e^{-x^2})

4.\:\:\lim _{x\to \infty}(2^x)

5.\:\:\lim _{x\to \infty}(\ln(\frac{1}{x}))

6.\:\:\lim _{x\to \infty}(\ln(x+1))

7.\:\:\lim _{x\to \infty}(\arctan(e^{x}))

8.\:\:\lim _{x\to \frac{π}{2}+}(e^{\tan(x)})

9.\:\:\lim _{x\to -\infty }((1.001)^{x})

10.\:\:\lim _{x\to \infty }(x^{\frac{3}{x}})

Limits

Advanced

Squeeze Theorem

1.\:\:\lim _{x\to \infty}(\frac{\cos(x)}{x})

2.\:\:\lim _{x\to 0}x^2cos(\frac{1}{x})

3.\:\:\lim _{x\to 0}(x\sin (\frac{1}{x}))

4.\:\:\lim _{x\to 0}(x^{2}\sin(\frac{1}{x}))

5.\:\:\lim _{x\to \infty}(\frac{\cos^{2}(x-4)}{2-x^{2}})

6.\:\:\lim _{x\to 0}(x^8\sin (\frac{1}{x}))

7.\:\:\lim _{x\to \infty}(2x+x\sin (x))

8.\:\:\lim _{x\to 0}(x^{2}e^{\sin (\frac{1}{x})})

9.\:\:\lim _{x\to 0}(x\sin (\frac{1}{x^{2}}))

10.\:\:\lim _{x\to \infty}(2\sin (\frac{x}{2})-3x)