Limits of functions : 11th grade math worksheets with answers.

11th grade math worksheets

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Math worksheets in 11th grade on limits and asymptotes.

Exercise 1 – Limits in infinity

Determine in each case $\lim_{x\,\mapsto \,+\infty\,}f(x)$ .

1. $f(x)=\frac{sinx+1}{2x}\;\,D=\mathbb{R}^*.$

2. $f(x)=2\sqrt{x}-sin(3x+1)\,;D=\mathbb{R}^+.$

Exercise 2 – Definition and limits

Determine the domain of definition D of f and then study the limits of f on the boundaries of D.

$f(x)=\sqrt{\frac{x-1}{4x+2}}$

Exercise 3 – Limit of a rational function

Determine the limit in $+\infty$ and $-\infty$ of :

$g(x)=\frac{3x^3+2x^2+1}{2x^4+3x^2-5}$

Exercise 4 – Calculate the following limits

$1.\,\lim_{x\,\mapsto \,0\,}\frac{1}{x^2}-\sqrt{x}\\2.\,\lim_{x\,\mapsto \,+\infty}\frac{1}{x^2}-\sqrt{x}\\3.\,\lim_{x\,\mapsto \,+\infty\,}(\frac{1}{x-2}-1)$

Exercise 5 – Functions, derivative and tangent

Let be the function defined on $\mathbb{R}$ by .

We note its graphical representation.

1. Calculate the derivative of , then solve the equation .

2. Deduce the coordinates of s two points A and B in which

has a horizontal tangent .

3. Determine the coordinates of the three points P, Q and R of intersection

between and the abscissa axis (P will be noted the one which has a strictly positive abscissa)

4. Deduce an equation of the tangent T at in P.

Exercise 6 – Functions, derivative and limit

Let be the function defined on $\mathbb{R}\setminus,,\{,2,,\}$ by $f(x)=\frac{2x-5}{x-2}$ .

1. Study the following limits:

$\lim_{x\to,+\infty,}f(x)$ and $\lim_{x\to,2,,x>,2}f(x)$ .

2. Calculate the derivative of . What is its sign?

Exercise 7 – Calculation of limits of functions

Determine the following limits:

1. $\lim_{x\,\to\,+\infty}\,-\frac{1}{x}\,$

2. $\lim_{x\,\to\,+\infty}\,-\frac{x^2}{2}\,$

3. $\lim_{x\,\to\,+\infty}\,2+\frac{1}{x}\,$

4. $\lim_{x\,\to\,+\infty}\,-5x^3\,$

Exercise 8

Determine the following limits:
1. $\lim_{x\,\to\,-\infty}\,-\frac{1}{x}\,$

2. $\lim_{x\,\to\,-\infty}\,-\frac{x^2}{2}\,$

3. $\lim_{x\,\to\,-\infty}\,2+\frac{1}{x}\,$

4. $\lim_{x\,\to\,-\infty}\,-5x^3\,$

5. $\lim_{x\,\to\,-\infty}\,-\sqrt{-x}\,$

6. $\lim_{x\,\to\,-\infty}\,-\frac{2}{-5x^3}\,$

Exercise 9

Determine the following limits:
1. $\lim_{x\,\to\,0^-}\,-\frac{1}{x}\,$

2. $\lim_{x\,\to\,{-3}^-}\,x+3\,$

3. $\lim_{x\,\to\,{2}^-}\,2-x^2\,$

4. $\lim_{x\,\to\,{3}^+}\,\frac{1}{2x-6}\,$

5. $\lim_{x\,\to\,{3}^+}\,\frac{-7}{6-2x}\,$

Exercise 10

Determine the following sum or difference limits:
1. $\lim_{x\,\to\,+\infty}\,x^2+3x+5\,$

2. $\lim_{x\,\to\,+\infty}\,10-3x-x^2\,$

3. $\lim_{x\,\to\,+\infty}\,3+\frac{1}{x}-\frac{6}{x^2}\,$

Exercise 11

Determine the limits of the following quotients and square roots:
1. $\lim_{x\,\to\,0^-}\,\frac{3x^2+6x-5}{x^3}\,$

2. $\lim_{x\,\to\,2^-}\,\frac{4x^2+2x-1}{x-2}\,$

3. $\lim_{x\,\to\,6^-}\,5x+6-\frac{7}{x-6}\,$

4. $\lim_{x\,\to\,{-3}^-}\,\frac{4-x^2}{(x+3)^2}\,$

5. $\lim_{x\,\to\,{6}^+}\,\sqrt{2x-12}\,$

6. $\lim_{x\,\to\,{3}^-}\,\sqrt{\frac{1}{3-x}}\,$

7. $\lim_{x\,\to\,{3}^+}\,5x+3-\frac{1}{\sqrt{3-x}}\,$

Exercise 12

Specify the limit of the following indeterminate forms:
1. $\lim_{x\,\to\,+\infty}\,x^2-3x+5\,$

2. $\lim_{x\,\to\,+\infty}\,1-x^2+x^3\,$

3. $\lim_{x\,\to\,-\infty}\,x^2+5x-1\,$

4. $\lim_{x\,\to\,+\infty}\,\frac{3x^2+5x-9}{5x^2-1}\,$

5. $\lim_{x\,\to\,-\infty}\,\frac{8x^6-9}{5x^4-3}\,$

6. $\lim_{x\,\to\,+\infty}\,x-\sqrt{x}\,$

7. $\lim_{x\,\to\,+\infty}\,\sqrt{x+2}-\sqrt{x}$

8. $\lim_{x\,\to\,-\infty}\,\sqrt{3-x}-\sqrt{4-x}$

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