Limit of a Function

Nerd Cafe

Example 1

• Limit:

$$\lim_{x \to 2} (3x+1)$$

• Solution:

$$3\times (2)+1=7$$

• Python Code:

from sympy import symbols, limit

x = symbols('x')
expr = 3*x + 1
result = limit(expr, x, 2)
print(result)

• Output:

7

Example 2

• Limit:

$$\lim_{x \to 0} \frac{sin(x)}{x}$$

• Solution: Using Taylor Series

Use the Taylor expansion of sin(x) around 0:

$$sin(x)=x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-...$$

So:

$$\frac{sin(x)}{x}=\frac{x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-...}{x}=1-\frac{x^{2}}{6}+\frac{x^{4}}{120}-...$$

Now take the limit:

$$\lim_{x \to 0} \left( 1-\frac{x^{2}}{6}+\frac{x^{4}}{120}-... \right)=1$$

• Python Code:

from sympy import sin

expr = sin(x)/x
result = limit(expr, x, 0)
print(result)

• Output:

1

Example 3

• Limit:

$$\lim_{x \to 1} \frac{x^{2}-1}{x-1}$$

• Solution: Factor numerator:

$$\frac{x^{2}-1}{x-1}=\frac{(x-1)(x+1)}{x-1}=x+1\Rightarrow \lim_{x \to 1} (x+1)=2$$

• Python Code:

expr = (x**2 - 1)/(x - 1)
result = limit(expr, x, 1)
print(result)

• Output:

2

Example 4

• Limit:

$$\lim_{x \to 0} \frac{1-cos(x)}{x^{2}}$$

• Solution: This is a known standard limit:

Expand cos⁡(𝑥) using Taylor series:

$$cos(x)=1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-...$$

So:

$$1-cos(x)=1-\left( 1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-... \right)=\frac{x^{2}}{2!}-\frac{x^{4}}{4!}+...$$

Substitute into the limit:

$$\lim_{x \to 0} \left( \frac{1-cos(x)}{x^{2}} \right)=\lim_{x \to 0} \left( \frac{\frac{x^{2}}{2!}-\frac{x^{4}}{4!}+...}{x^{2}} \right)$$

Simplify the expression:

$$\lim_{x \to 0} \left( \frac{1}{2!}-\frac{x^{2}}{4!}+... \right)=\frac{1}{2}$$

• Python Code:

from sympy import cos

expr = (1 - cos(x))/x**2
result = limit(expr, x, 0)
print(result)

• Output:

1/2

Example 5

• Limit:

$$\lim_{x \to \infty } \frac{5x^{2}+5}{2x^{2}+7}$$

• Solution:

$$\lim_{x \to \infty } \frac{5x^{2}+5}{2x^{2}+7}=\lim_{x \to \infty } \frac{5+\frac{5}{x^{2}}}{2+\frac{7}{x^{2}}}=\frac{5}{2}$$

• Python Code:

from sympy import oo

expr = (5*x**2 + 3)/(2*x**2 + 7)
result = limit(expr, x, oo)
print(result)

• Output

5/2

Example 6

• Limit:

$$\lim_{x \to 0} \frac{x}{\left| x \right|}$$

• Solution:

  • From left: −1

  • From right: 1

  • Left ≠ Right → Limit does not exist

• Python Code:

from sympy import Abs

expr = x / Abs(x)
left_limit = limit(expr, x, 0, dir='-')
right_limit = limit(expr, x, 0, dir='+')
print(f"Left: {left_limit}, Right: {right_limit}")

• Output

Left: -1, Right: 1

Keywords

limit, calculus, trigonometric limits, Taylor series, squeeze theorem, sin(x)/x, fundamental limit, L'Hôpital's rule, limit definition, mathematical proof, nerd cafe