Problem 13 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \left.\lim _{x \rightarrow 1} 3 x^{2}+4 x+5 . \text { \\{Ans. } 12\right\\} $$

Step-by-Step Solution

Answer

The limit of the function \(f(x) = 3x^2 + 4x + 5\) as \(x\) approaches 1 is 12, as found by evaluating the function at \(x=1\): \(\lim_{x \rightarrow 1}(3x^2 + 4x + 5) = 12\).

Step 1: Understanding the given function

We are given the quadratic function \(f(x) = 3x^2 + 4x + 5\). We need to find the limit of this function as \(x\) approaches 1. Let's move to the next step to evaluate this limit.

Step 2: Evaluating the limit of the function

To find the limit of the function as \(x\) approaches 1, we need to substitute the value of \(x\) in the given function and simplify it: \[ \lim_{x \rightarrow 1}(3x^2 + 4x + 5) \] Now we substitute \(x=1\): \[ f(1) = 3(1)^2 + 4(1) + 5 \] Simplify the expression: \[ f(1) = 3 + 4 + 5 = 12 \]

Step 3: Writing the answer

The limit of the given function \(3x^2 + 4x + 5\) as \(x\) approaches 1 is equal to 12. So, the final answer is: \(\lim_{x \rightarrow 1}(3x^2 + 4x + 5) = 12\).

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function that can be expressed in the standard form of \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). This particular function appears as a parabola on a graph, either opening upwards or downwards depending on the sign of \(a\).

In our case, the quadratic function given is \(3x^2 + 4x + 5\). Here, \(a = 3\), \(b = 4\), and \(c = 5\). Each part of the equation has a specific role:

• The term \(3x^2\) defines the primary parabolic shape and its direction due to the positive coefficient.
• The term \(4x\) represents the linear component, which adjusts the slope of the parabola.
• The constant \(5\) shifts the graph vertically upwards by 5 units.

Understanding these components helps in analyzing the overall behavior of the quadratic function. Knowing how to evaluate and graph this function assists in solving many calculus problems, including limit evaluation.

Substitution Method

The substitution method is used for evaluating limits by directly substituting the value into the function wherever the variable is found, mainly when the function is continuous at that point.

For the limit \(\lim_{x \rightarrow 1}(3x^2 + 4x + 5)\), we can substitute \(x = 1\) since the polynomial is continuous at \(x = 1\). Here's a breakdown of how substitution works:

• Identify the variable in the function: In this function, it is \(x\).
• Replace \(x\) with the approaching value, which is 1: Substitute 1 into \(3x^2 + 4x + 5\).
• Simplify the expression: Plugging in 1 gives us \(3(1)^2 + 4(1) + 5\), which simplifies to 12.

This method is efficient when dealing with polynomial functions like quadratics, as they are continuous across all real numbers. Thus, direct substitution is straightforward.

Limit Evaluation

Limit evaluation is an important concept in calculus, used to find the value that a function approaches as the input approaches a certain point. It's crucial in understanding the behavior of functions around specific points, especially where the function might not be well defined.

For the given problem, we want to evaluate the limit of the function \(3x^2 + 4x + 5\) as \(x\) approaches 1. When evaluating limits of simple polynomial functions, especially quadratics, substitution often provides an immediate solution. Since these functions are continuous, substituting the value of \(x\) gives the limit.

Here, performing \(\lim_{x \rightarrow 1}(3x^2 + 4x + 5)\), we substitute \(x = 1\) and find \(3(1)^2 + 4 \times 1 + 5 = 12\).

Limit evaluation is not only a vital tool for solving problems but also helps in understanding the approach behavior of different kinds of functions. Practicing limit evaluation on simple quadratic functions like this develops a solid base for tackling more complex calculus problems.