# Limit at an undefined point

Instructions: 1. Enter any function you like in the 2 text boxs. (e.g. $\frac{x^2}{x}$) The upper box is for the numerator. (e.g. x^2) The lower box is for the denominator. (e.g. x) 2. Uncheck 'enter function' after entering the function to check if your input has been successfully recognized. 3. Hold down 'shift' and drag to adjust the view if necessary. You can also adjust the scale by holding down 'shift' and drag the axis. 4. The point is undefined when the denominator equals zero. The undefined point will be marked by a white 'hole'. 5. Drag the slider or enter any value from -10 to 10 in the box, such that the point move closer and closer towards the undefined point. Observe how the limit changes. Notice that unless the red point is on the undefined point, the limit exactly equals the value of the function at the point. At the undefined point, the function at this point is not defined, but the limit at this point exists. 6. Repeat steps 1 to 8 if necessary. *Due to technical difficulties, I wasn't able to create more than one hole to represent a function with more than one undefined point.

Tasks: For L2: Try the following functions: 1. x(x-1)/(x-1) 2. x^3/x For L5: Try the following: 1. Write down the derivative of sin(x) when x equals 1 using limit, and in terms of h. Answer: $\frac{sin(1+h)-sin(1)}{h}$ 2. Replace h by x and enter it into the input f(x) box. Answer: $\frac{sin(1+x)-sin(1)}{x}$ 3. Drag the slider to the point at x equals 0, and observe how the limit changes. The limit of the function at x equals 0 is equivalent to the derivative of sin(x) when x equals 1. 4. Repeat steps 1 to 3 for any function you like. Recommended functions: 1. cos(x) 2. tan(x) 3. e^x 4. ln(x) Created for SCNC1111 Calculus Tutorials Applicable topics: L2. Limit at an undefined point L5. First principles