Find Limits of Functions in Calculus
Find the limits of various functions using different methods. Several Examples with detailed solutions are presented. More exercises with answers are at the end of this page.
Examples with Detailed Solutions
Example 1
Find the limit![limit of function in example 1](http://www.analyzemath.com/calculus/limits/limit_example_1.png)
Solution to Example 1:
Note that we are looking for the limit as x approaches 1 from the left ( x → 1- means x approaches 1 by values smaller than 1). Hence
x < 1
x - 1 < 0
If x - 1 < 0 then
| x - 1 | = - (x - 1)
Simplify to obtain
Example 2
Evaluate the limit![limit example 2](http://www.analyzemath.com/calculus/limits/find-limit-2-1.png)
Although the limit in question is the ratio of two polynomials, x = 5 makes both the numerator and denominator equal to zero. We need to factor both numerator and denominator as shown below.
![limit example 2 step 1](http://www.analyzemath.com/calculus/limits/find-limit-2-2.png)
Simplify to obtain
![limit example 2 step 2](http://www.analyzemath.com/calculus/limits/find-limit-2-3.png)
Example 3
Determine the limit![limit example 3](http://www.analyzemath.com/calculus/limits/find_limit_3_1.gif)
We need to look at the limit from the left of 2 and the limit from the right of 2. As x approaches 2 from the left x - 2 < 0 hence
|x - 2| = - (x - 2)
Substitute to obtain the limit from the left of 2 as follows
![limit solution to example 3, smaller values](http://www.analyzemath.com/calculus/limits/find_limit_3_2.gif)
As x approaches 2 from the right x - 2 > 0 hence
|x - 2| = x - 2
Substitute to obtain the limit from the right of 2 as follows
![final limit solution to example 3, larger values](http://www.analyzemath.com/calculus/limits/find_limit_3_3.gif)
Example 4
Calculate the limit![limit example 4](http://www.analyzemath.com/calculus/limits/find_limit_4_1.gif)
As x approaches -1, cube root x + 1 approaches 0 and ln(x+1) approaches - infinity hence an indeterminate form 0 × infinity
![limit solution to example 4, first step](http://www.analyzemath.com/calculus/limits/find_limit_4_2.gif)
Let us rewrite the limit so that it is of the infinity/infinity indeterminate form.
![limit solution to example 4, second step](http://www.analyzemath.com/calculus/limits/find_limit_4_3.gif)
We now use L'hopital's Rule and find the limit.
![limit solution to example 4, last step](http://www.analyzemath.com/calculus/limits/find_limit_4_4.gif)
Example 5
Find the limit![limit example 5](http://www.analyzemath.com/calculus/limits/find_limit_5_1.gif)
Solution to Example 5:
As x gets larger x + 1 gets larger, 1/(x+1) approaches zero, e^(1/(x+1)) approaches 1 and e^(1/(x+1)) - 1 approaches 0 hence an indeterminate form: ∞ × 0
![limit solution to example 5, first step](http://www.analyzemath.com/calculus/limits/find_limit_5_2.gif)
Let us rewrite the limit so that it is of the indeterminate form 0/0.
![limit solution to example 5, second step](http://www.analyzemath.com/calculus/limits/find_limit_5_3.gif)
Apply the l'hopital's theorem to find the limit.
![limit solution to example 5, last step](http://www.analyzemath.com/calculus/limits/find_limit_5_4.gif)
Example 6
Calculate the limit![limit example 6](http://www.analyzemath.com/calculus/limits/find_limit_6_1.gif)
Solution to Example 6:
As x approaches 9, both numerator and denominator approach 0. Multiply both numerator and denominator by the conjugate of the numerator.
Expand and simplify.
and now find the limit.
Example 7
Find the limit![limit example 7](http://www.analyzemath.com/calculus/limits/find_limit_7_1.gif)
Solution to Example 7:
The range of the cosine function is.
-1 ≤ cos x ≤ 1
Divide all terms of the above inequality by x, for x positive.
-1 / x ≤ cos x / x ≤ 1 / x
Now as x takes larger values without bound (+infinity) both -1 / x and 1 / x approaches 0. Hence by the squeezing theorem the above limit is given by
![limit solution to example 7](http://www.analyzemath.com/calculus/limits/find_limit_7_2.gif)
Example 8
Find the limit![limit example 8](http://www.analyzemath.com/calculus/limits/find_limit_8_1.gif)
Solution to Example 8:
As t approaches 0, both the numerator and denominator approach 0 and we have the 0 / 0 indeterminate form. Hence the l'hopital theorem is used to calculate the above limit as follows
![limit solution to example 8](http://www.analyzemath.com/calculus/limits/find_limit_8_2.gif)
Example 9
Calculate the limit![limit example 9](http://www.analyzemath.com/calculus/limits/find_limit_9_1.gif)
Solution to Example 9:
We first factor out 16 x 2 under the square root of the denominator and take out of the square root and rewrite the limit as
![limit solution to example 9, first step](http://www.analyzemath.com/calculus/limits/find_limit_9_2.gif)
![limit solution to example 9, second step](http://www.analyzemath.com/calculus/limits/find_limit_9_3.gif)
Example 10
Find the limit![limit example 10](http://www.analyzemath.com/calculus/limits/find_limit_10_1.gif)
Solution to Example 10:
As x approaches 2 from the left then x - 2 approaches 0 from the left or x - 2 < 0. The numerator approaches 5 and the denominator approaches 0 from the left hence the limit is given by
![limit solution to example 10](http://www.analyzemath.com/calculus/limits/find_limit_10_2.gif)
Example 11
Calculate the limit![limit example 11](http://www.analyzemath.com/calculus/limits/find_limit_11_1.gif)
Solution to Example 11:
Factor x 2 in the denominator and simplify.
![limit solution to example 11](http://www.analyzemath.com/calculus/limits/find_limit_11_2.gif)
As x takes large values (infinity), the terms 2/x and 1/x 2 approaches 0 hence the limit is
Example 12
Find the limit![limit example 12](http://www.analyzemath.com/calculus/limits/find_limit_12_1.gif)
Solution to Example 12:
Factor x 2 in the numerator and denominator and simplify.
![limit solution to example 12](http://www.analyzemath.com/calculus/limits/find_limit_12_2.gif)
As x takes large values (infinity), the terms 1/x and 1/x 2 and 3/x 2 approaches 0 hence the limit is
Example 13
Determine the limit![limit example 13](http://www.analyzemath.com/calculus/limits/find_limit_13_1.gif)
Solution to Example 13:
Multiply numerator and denominator by 3t.
![limit solution to example 13, step 1](http://www.analyzemath.com/calculus/limits/find_limit_13_2.gif)
Use limit properties and theorems to rewrite the above limit as the product of two limits and a constant.
![limit solution to example 13, step 2](http://www.analyzemath.com/calculus/limits/find_limit_13_3.gif)
![limit solution to example 13, step 3](http://www.analyzemath.com/calculus/limits/find_limit_13_4.gif)
![limit solution to example 13, step 4](http://www.analyzemath.com/calculus/limits/find_limit_13_5.gif)
![limit solution to example 13, step 5](http://www.analyzemath.com/calculus/limits/find_limit_13_6.gif)
Example 14
Find the limit![limit example 14](http://www.analyzemath.com/calculus/limits/find_limit_14_1.gif)
Solution to Example 14:
Factor x 2 inside the square root and use the fact that sqrt(x 2 ) = | x |.
![limit solution to example 14, step 1](http://www.analyzemath.com/calculus/limits/find_limit_14_2.gif)
Since x takes large values (infinity) then | x | = x. Hence the indeterminate form
![limit solution to example 14, step 2](http://www.analyzemath.com/calculus/limits/find_limit_14_3.gif)
Multiply numerator and denominator by the conjugate and simplify
![limit solution to example 14, step 3](http://www.analyzemath.com/calculus/limits/find_limit_14_4.gif)
![limit solution to example 14, step 4](http://www.analyzemath.com/calculus/limits/find_limit_14_5.gif)
Example 15
Determine the limit![limit example 15](http://www.analyzemath.com/calculus/limits/find_limit_15_1.gif)
Solution to Example 15:
Let z = 1 / x so that as x get large x approaches 0. Substitute and calculate the limit as follows.
![limit solution to example 12](http://www.analyzemath.com/calculus/limits/find_limit_15_2.gif)
Exercises
Calculate the following limits1)
![Calculate limit question 1](http://www.analyzemath.com/calculus/limits/question_1.gif)
2)
![Calculate limit question 2](http://www.analyzemath.com/calculus/limits/question_2.gif)
3)
![Calculate limit question 3](http://www.analyzemath.com/calculus/limits/question_3.gif)
4)
![Calculate limit question 4](http://www.analyzemath.com/calculus/limits/question_4.gif)
5)
![Calculate limit question 5](http://www.analyzemath.com/calculus/limits/question_5.gif)
6)
![Calculate limit question 6](http://www.analyzemath.com/calculus/limits/question_6.gif)
Answers to Above Exercises
1) 3
2) 1
3) 1
4) 1/4
5) 0
6) 4
More References and links
Calculus Tutorials and ProblemsLimits of Absolute Value Functions Questions
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