Group 01Group Members:-

1. Umer Rashid (14093122-002)2. Adnan Aslam (14093122-003)3. Bilal Amjad (4093122-004)

Limits of complex functions1st definitionLet f be a function of z. To say that 

means that f (z) can be made arbitrarily close to if we choose the point z close enough to but distinct from it. We now express the definition of limit in a precise and usable form.How close is "close enough to " depends on how close one wants to make f(z) to .

2nd definition

This definition is also known as "epsilon-delta definition of limit“. It was first given by Bernard Bolzano in 1817.For a positive number ε, there is a positive number δ such that |f (z) − | < ε whenever 0 < |z − | < δ.Therefore let the positive number ε (epsilon) be how close one wishes to make f(z) to strictly one wants the distance to be less than ε. Further, if the positive number δ is how close one will make z to , and if the distance from z to is less than δ (but not zero), then the distance from f(z) to will be less than ε. Therefore δ depends on ε. The limit statement means that no matter how small ε is made, δ can be made small enough.

• The limit statement means that no matter how small ε is made, δ can be made small enough.

• The letters ε and δ can be as "error" and "distance", and in fact Cauchy used ε as an abbreviation for "error" in some of his work. In these terms, the error (ε) in the measurement of the value at the limit can be made as small as desired by reducing the distance (δ) to the limit point.

lim𝑧→ 3 𝑖

𝑧 2+9𝑧−3 𝑖=6 𝑖

So whenever Hence limit exist

Example 1:-

Example 2:-Take the limit of the f(x) as x approaches

f(0.9) f(0.99) f(0.999)

f() f(1.001)

f(1.01) f(1.1)

1.900 1.990 1.999 undefined

2.001 2.010 2.100

As x approaches from both sides f(x) gets closer and closer to 2, therefore making 2 the limit of f(x) as x approaches . The limit still exists even though f() does not exist because as x draws closer to f(x) draws closer to 2 making it the limit to the function.

Civil EngineeringMany aspects of civil engineering require calculus. Firstly, derivation of the

basic fluid mechanics equations requires limits. For example, all hydraulic analysis programs, which aid in the design of storm drain and open channel systems, use calculus numerical methods to obtain the results.

Some applications of limit:-

Mechanical Engineering In mechanical engineering, limit is used for computing the surface area of

complex objects to determine frictional forces, designing a pump according to flow rate and head, and calculating the power provided by a battery system

Aerospace EngineeringAnalysis of rockets that function in stages require calculus, as does

gravitational modeling over time and space. Almost all physics models, especially those of astronomy and complex systems, use some form of limits.

The reading of your speedometer (e.g., 85 km/h) is a limit in the real world. Your speed is changing continuously during time.

BusinessIn business, limits can help us by

providing an accurate and measurable way to record changes in variables using numbers and mathematics.

Derivatives in calculus can be used to determine maximum profit, minimum cost, rate of change of cost, and how to maximize/ minimize profit, cost or production.

MedicineBy using the principles of calculus, we can find how fast a tumor

is shrinking/growing, the size when the tumor will stop growing and when certain treatments should be given and finding the volume of a tumor.

Calculus can be used to determine the blood flow in an artery or a vein at a given point in time.

Calculus can be used to find the amount of blood pumped through the heart per unit time.

In biology, it is utilized to formulate rates such as birth and death rates.

• To a machinist manufacturing a piston, close may mean within a few thousandths of an inch.

• To an astronomer studying distant galaxies, close may mean within a few thousand light-years.

Thank you